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G = C7×C22.36C24  order 448 = 26·7

Direct product of C7 and C22.36C24

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Aliases: C7×C22.36C24, C14.1152- (1+4), C14.1572+ (1+4), C4⋊Q8⋊11C14, (D4×C28)⋊42C2, (C4×D4)⋊13C14, (C4×Q8)⋊10C14, (Q8×C28)⋊30C2, C22⋊Q8⋊9C14, C4⋊D4.9C14, C4.4D4⋊9C14, C42⋊2C2⋊3C14, C42.40(C2×C14), C42⋊C2⋊13C14, C28.278(C4○D4), (C4×C28).281C22, (C2×C14).362C24, (C2×C28).671C23, C22.D4⋊7C14, C2.7(C7×2- (1+4)), C2.9(C7×2+ (1+4)), (D4×C14).219C22, (C22×C14).97C23, C22.36(C23×C14), C23.14(C22×C14), (Q8×C14).182C22, (C22×C28).450C22, (C7×C4⋊Q8)⋊32C2, C4.22(C7×C4○D4), C4⋊C4.70(C2×C14), C2.19(C14×C4○D4), (C7×C22⋊Q8)⋊36C2, (C2×D4).33(C2×C14), C14.238(C2×C4○D4), (C7×C4.4D4)⋊29C2, (C7×C4⋊D4).19C2, C22⋊C4.4(C2×C14), (C2×Q8).26(C2×C14), (C7×C42⋊2C2)⋊14C2, (C7×C42⋊C2)⋊34C2, (C7×C4⋊C4).249C22, (C2×C4).29(C22×C14), (C22×C4).62(C2×C14), (C7×C22.D4)⋊26C2, (C7×C22⋊C4).150C22, SmallGroup(448,1325)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.36C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C4.4D4 — C7×C22.36C24
Lower central
C1 — C22 — C7×C22.36C24
Upper central
C1 — C2×C14 — C7×C22.36C24

Subgroups: 322 in 216 conjugacy classes, 146 normal (62 characteristic)
C1, C2 [×3], C2 [×3], C4 [×2], C4 [×11], C22, C22 [×9], C7, C2×C4 [×6], C2×C4 [×6], C2×C4 [×4], D4 [×4], Q8 [×4], C23, C23 [×2], C14 [×3], C14 [×3], C42 [×2], C42 [×2], C22⋊C4 [×2], C22⋊C4 [×10], C4⋊C4 [×4], C4⋊C4 [×6], C22×C4, C22×C4 [×2], C2×D4, C2×D4 [×2], C2×Q8, C2×Q8 [×2], C28 [×2], C28 [×11], C2×C14, C2×C14 [×9], C42⋊C2, C4×D4, C4×Q8, C4⋊D4, C22⋊Q8, C22⋊Q8 [×2], C22.D4 [×2], C4.4D4, C4.4D4 [×2], C42⋊2C2 [×2], C4⋊Q8, C2×C28 [×6], C2×C28 [×6], C2×C28 [×4], C7×D4 [×4], C7×Q8 [×4], C22×C14, C22×C14 [×2], C22.36C24, C4×C28 [×2], C4×C28 [×2], C7×C22⋊C4 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×4], C7×C4⋊C4 [×6], C22×C28, C22×C28 [×2], D4×C14, D4×C14 [×2], Q8×C14, Q8×C14 [×2], C7×C42⋊C2, D4×C28, Q8×C28, C7×C4⋊D4, C7×C22⋊Q8, C7×C22⋊Q8 [×2], C7×C22.D4 [×2], C7×C4.4D4, C7×C4.4D4 [×2], C7×C42⋊2C2 [×2], C7×C4⋊Q8, C7×C22.36C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×2], C24, C2×C14 [×35], C2×C4○D4, 2+ (1+4), 2- (1+4), C22×C14 [×15], C22.36C24, C7×C4○D4 [×2], C23×C14, C14×C4○D4, C7×2+ (1+4), C7×2- (1+4), C7×C22.36C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=f2=1, e2=cb=bc, g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=gdg-1=bd=db, fef=be=eb, bf=fb, bg=gb, fdf=cd=dc, ce=ec, cf=fc, cg=gc, eg=ge, fg=gf >

Smallest permutation representation
►On 224 points
Generators in S224
(1 2 3 4 5 6 7)(8 9 10 11 12 13 14)(15 16 17 18 19 20 21)(22 23 24 25 26 27 28)(29 30 31 32 33 34 35)(36 37 38 39 40 41 42)(43 44 45 46 47 48 49)(50 51 52 53 54 55 56)(57 58 59 60 61 62 63)(64 65 66 67 68 69 70)(71 72 73 74 75 76 77)(78 79 80 81 82 83 84)(85 86 87 88 89 90 91)(92 93 94 95 96 97 98)(99 100 101 102 103 104 105)(106 107 108 109 110 111 112)(113 114 115 116 117 118 119)(120 121 122 123 124 125 126)(127 128 129 130 131 132 133)(134 135 136 137 138 139 140)(141 142 143 144 145 146 147)(148 149 150 151 152 153 154)(155 156 157 158 159 160 161)(162 163 164 165 166 167 168)(169 170 171 172 173 174 175)(176 177 178 179 180 181 182)(183 184 185 186 187 188 189)(190 191 192 193 194 195 196)(197 198 199 200 201 202 203)(204 205 206 207 208 209 210)(211 212 213 214 215 216 217)(218 219 220 221 222 223 224)

(1 34)(2 35)(3 29)(4 30)(5 31)(6 32)(7 33)(8 219)(9 220)(10 221)(11 222)(12 223)(13 224)(14 218)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 46)(37 47)(38 48)(39 49)(40 43)(41 44)(42 45)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 64)(58 65)(59 66)(60 67)(61 68)(62 69)(63 70)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 148)(142 149)(143 150)(144 151)(145 152)(146 153)(147 154)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 176)(170 177)(171 178)(172 179)(173 180)(174 181)(175 182)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 204)(198 205)(199 206)(200 207)(201 208)(202 209)(203 210)

(1 45)(2 46)(3 47)(4 48)(5 49)(6 43)(7 44)(8 18)(9 19)(10 20)(11 21)(12 15)(13 16)(14 17)(22 223)(23 224)(24 218)(25 219)(26 220)(27 221)(28 222)(29 37)(30 38)(31 39)(32 40)(33 41)(34 42)(35 36)(50 70)(51 64)(52 65)(53 66)(54 67)(55 68)(56 69)(57 75)(58 76)(59 77)(60 71)(61 72)(62 73)(63 74)(78 98)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 103)(86 104)(87 105)(88 99)(89 100)(90 101)(91 102)(106 126)(107 120)(108 121)(109 122)(110 123)(111 124)(112 125)(113 131)(114 132)(115 133)(116 127)(117 128)(118 129)(119 130)(134 154)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 210)(191 204)(192 205)(193 206)(194 207)(195 208)(196 209)(197 215)(198 216)(199 217)(200 211)(201 212)(202 213)(203 214)

(1 158)(2 159)(3 160)(4 161)(5 155)(6 156)(7 157)(8 131)(9 132)(10 133)(11 127)(12 128)(13 129)(14 130)(15 117)(16 118)(17 119)(18 113)(19 114)(20 115)(21 116)(22 124)(23 125)(24 126)(25 120)(26 121)(27 122)(28 123)(29 136)(30 137)(31 138)(32 139)(33 140)(34 134)(35 135)(36 148)(37 149)(38 150)(39 151)(40 152)(41 153)(42 154)(43 145)(44 146)(45 147)(46 141)(47 142)(48 143)(49 144)(50 162)(51 163)(52 164)(53 165)(54 166)(55 167)(56 168)(57 169)(58 170)(59 171)(60 172)(61 173)(62 174)(63 175)(64 176)(65 177)(66 178)(67 179)(68 180)(69 181)(70 182)(71 183)(72 184)(73 185)(74 186)(75 187)(76 188)(77 189)(78 190)(79 191)(80 192)(81 193)(82 194)(83 195)(84 196)(85 197)(86 198)(87 199)(88 200)(89 201)(90 202)(91 203)(92 204)(93 205)(94 206)(95 207)(96 208)(97 209)(98 210)(99 211)(100 212)(101 213)(102 214)(103 215)(104 216)(105 217)(106 218)(107 219)(108 220)(109 221)(110 222)(111 223)(112 224)

(1 98 42 102)(2 92 36 103)(3 93 37 104)(4 94 38 105)(5 95 39 99)(6 96 40 100)(7 97 41 101)(8 163 25 169)(9 164 26 170)(10 165 27 171)(11 166 28 172)(12 167 22 173)(13 168 23 174)(14 162 24 175)(15 180 223 184)(16 181 224 185)(17 182 218 186)(18 176 219 187)(19 177 220 188)(20 178 221 189)(21 179 222 183)(29 86 47 80)(30 87 48 81)(31 88 49 82)(32 89 43 83)(33 90 44 84)(34 91 45 78)(35 85 46 79)(50 119 63 106)(51 113 57 107)(52 114 58 108)(53 115 59 109)(54 116 60 110)(55 117 61 111)(56 118 62 112)(64 131 75 120)(65 132 76 121)(66 133 77 122)(67 127 71 123)(68 128 72 124)(69 129 73 125)(70 130 74 126)(134 210 147 214)(135 204 141 215)(136 205 142 216)(137 206 143 217)(138 207 144 211)(139 208 145 212)(140 209 146 213)(148 191 159 197)(149 192 160 198)(150 193 161 199)(151 194 155 200)(152 195 156 201)(153 196 157 202)(154 190 158 203)

(8 25)(9 26)(10 27)(11 28)(12 22)(13 23)(14 24)(15 223)(16 224)(17 218)(18 219)(19 220)(20 221)(21 222)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 92)(86 93)(87 94)(88 95)(89 96)(90 97)(91 98)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 120)(114 121)(115 122)(116 123)(117 124)(118 125)(119 126)(134 154)(135 148)(136 149)(137 150)(138 151)(139 152)(140 153)(141 159)(142 160)(143 161)(144 155)(145 156)(146 157)(147 158)(162 182)(163 176)(164 177)(165 178)(166 179)(167 180)(168 181)(169 187)(170 188)(171 189)(172 183)(173 184)(174 185)(175 186)(190 203)(191 197)(192 198)(193 199)(194 200)(195 201)(196 202)(204 215)(205 216)(206 217)(207 211)(208 212)(209 213)(210 214)

(1 50 34 74)(2 51 35 75)(3 52 29 76)(4 53 30 77)(5 54 31 71)(6 55 32 72)(7 56 33 73)(8 191 219 215)(9 192 220 216)(10 193 221 217)(11 194 222 211)(12 195 223 212)(13 196 224 213)(14 190 218 214)(15 208 22 201)(16 209 23 202)(17 210 24 203)(18 204 25 197)(19 205 26 198)(20 206 27 199)(21 207 28 200)(36 57 46 64)(37 58 47 65)(38 59 48 66)(39 60 49 67)(40 61 43 68)(41 62 44 69)(42 63 45 70)(78 130 102 106)(79 131 103 107)(80 132 104 108)(81 133 105 109)(82 127 99 110)(83 128 100 111)(84 129 101 112)(85 120 92 113)(86 121 93 114)(87 122 94 115)(88 123 95 116)(89 124 96 117)(90 125 97 118)(91 126 98 119)(134 162 158 186)(135 163 159 187)(136 164 160 188)(137 165 161 189)(138 166 155 183)(139 167 156 184)(140 168 157 185)(141 169 148 176)(142 170 149 177)(143 171 150 178)(144 172 151 179)(145 173 152 180)(146 174 153 181)(147 175 154 182)
 
G:=sub<Sym(224)| (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,218)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,131)(9,132)(10,133)(11,127)(12,128)(13,129)(14,130)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,124)(23,125)(24,126)(25,120)(26,121)(27,122)(28,123)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,145)(44,146)(45,147)(46,141)(47,142)(48,143)(49,144)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,98,42,102)(2,92,36,103)(3,93,37,104)(4,94,38,105)(5,95,39,99)(6,96,40,100)(7,97,41,101)(8,163,25,169)(9,164,26,170)(10,165,27,171)(11,166,28,172)(12,167,22,173)(13,168,23,174)(14,162,24,175)(15,180,223,184)(16,181,224,185)(17,182,218,186)(18,176,219,187)(19,177,220,188)(20,178,221,189)(21,179,222,183)(29,86,47,80)(30,87,48,81)(31,88,49,82)(32,89,43,83)(33,90,44,84)(34,91,45,78)(35,85,46,79)(50,119,63,106)(51,113,57,107)(52,114,58,108)(53,115,59,109)(54,116,60,110)(55,117,61,111)(56,118,62,112)(64,131,75,120)(65,132,76,121)(66,133,77,122)(67,127,71,123)(68,128,72,124)(69,129,73,125)(70,130,74,126)(134,210,147,214)(135,204,141,215)(136,205,142,216)(137,206,143,217)(138,207,144,211)(139,208,145,212)(140,209,146,213)(148,191,159,197)(149,192,160,198)(150,193,161,199)(151,194,155,200)(152,195,156,201)(153,196,157,202)(154,190,158,203), (8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,215)(205,216)(206,217)(207,211)(208,212)(209,213)(210,214), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,120,92,113)(86,121,93,114)(87,122,94,115)(88,123,95,116)(89,124,96,117)(90,125,97,118)(91,126,98,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182)>;
 
G:=Group( (1,2,3,4,5,6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21)(22,23,24,25,26,27,28)(29,30,31,32,33,34,35)(36,37,38,39,40,41,42)(43,44,45,46,47,48,49)(50,51,52,53,54,55,56)(57,58,59,60,61,62,63)(64,65,66,67,68,69,70)(71,72,73,74,75,76,77)(78,79,80,81,82,83,84)(85,86,87,88,89,90,91)(92,93,94,95,96,97,98)(99,100,101,102,103,104,105)(106,107,108,109,110,111,112)(113,114,115,116,117,118,119)(120,121,122,123,124,125,126)(127,128,129,130,131,132,133)(134,135,136,137,138,139,140)(141,142,143,144,145,146,147)(148,149,150,151,152,153,154)(155,156,157,158,159,160,161)(162,163,164,165,166,167,168)(169,170,171,172,173,174,175)(176,177,178,179,180,181,182)(183,184,185,186,187,188,189)(190,191,192,193,194,195,196)(197,198,199,200,201,202,203)(204,205,206,207,208,209,210)(211,212,213,214,215,216,217)(218,219,220,221,222,223,224), (1,34)(2,35)(3,29)(4,30)(5,31)(6,32)(7,33)(8,219)(9,220)(10,221)(11,222)(12,223)(13,224)(14,218)(15,22)(16,23)(17,24)(18,25)(19,26)(20,27)(21,28)(36,46)(37,47)(38,48)(39,49)(40,43)(41,44)(42,45)(50,74)(51,75)(52,76)(53,77)(54,71)(55,72)(56,73)(57,64)(58,65)(59,66)(60,67)(61,68)(62,69)(63,70)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,158)(135,159)(136,160)(137,161)(138,155)(139,156)(140,157)(141,148)(142,149)(143,150)(144,151)(145,152)(146,153)(147,154)(162,186)(163,187)(164,188)(165,189)(166,183)(167,184)(168,185)(169,176)(170,177)(171,178)(172,179)(173,180)(174,181)(175,182)(190,214)(191,215)(192,216)(193,217)(194,211)(195,212)(196,213)(197,204)(198,205)(199,206)(200,207)(201,208)(202,209)(203,210), (1,45)(2,46)(3,47)(4,48)(5,49)(6,43)(7,44)(8,18)(9,19)(10,20)(11,21)(12,15)(13,16)(14,17)(22,223)(23,224)(24,218)(25,219)(26,220)(27,221)(28,222)(29,37)(30,38)(31,39)(32,40)(33,41)(34,42)(35,36)(50,70)(51,64)(52,65)(53,66)(54,67)(55,68)(56,69)(57,75)(58,76)(59,77)(60,71)(61,72)(62,73)(63,74)(78,98)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,103)(86,104)(87,105)(88,99)(89,100)(90,101)(91,102)(106,126)(107,120)(108,121)(109,122)(110,123)(111,124)(112,125)(113,131)(114,132)(115,133)(116,127)(117,128)(118,129)(119,130)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,210)(191,204)(192,205)(193,206)(194,207)(195,208)(196,209)(197,215)(198,216)(199,217)(200,211)(201,212)(202,213)(203,214), (1,158)(2,159)(3,160)(4,161)(5,155)(6,156)(7,157)(8,131)(9,132)(10,133)(11,127)(12,128)(13,129)(14,130)(15,117)(16,118)(17,119)(18,113)(19,114)(20,115)(21,116)(22,124)(23,125)(24,126)(25,120)(26,121)(27,122)(28,123)(29,136)(30,137)(31,138)(32,139)(33,140)(34,134)(35,135)(36,148)(37,149)(38,150)(39,151)(40,152)(41,153)(42,154)(43,145)(44,146)(45,147)(46,141)(47,142)(48,143)(49,144)(50,162)(51,163)(52,164)(53,165)(54,166)(55,167)(56,168)(57,169)(58,170)(59,171)(60,172)(61,173)(62,174)(63,175)(64,176)(65,177)(66,178)(67,179)(68,180)(69,181)(70,182)(71,183)(72,184)(73,185)(74,186)(75,187)(76,188)(77,189)(78,190)(79,191)(80,192)(81,193)(82,194)(83,195)(84,196)(85,197)(86,198)(87,199)(88,200)(89,201)(90,202)(91,203)(92,204)(93,205)(94,206)(95,207)(96,208)(97,209)(98,210)(99,211)(100,212)(101,213)(102,214)(103,215)(104,216)(105,217)(106,218)(107,219)(108,220)(109,221)(110,222)(111,223)(112,224), (1,98,42,102)(2,92,36,103)(3,93,37,104)(4,94,38,105)(5,95,39,99)(6,96,40,100)(7,97,41,101)(8,163,25,169)(9,164,26,170)(10,165,27,171)(11,166,28,172)(12,167,22,173)(13,168,23,174)(14,162,24,175)(15,180,223,184)(16,181,224,185)(17,182,218,186)(18,176,219,187)(19,177,220,188)(20,178,221,189)(21,179,222,183)(29,86,47,80)(30,87,48,81)(31,88,49,82)(32,89,43,83)(33,90,44,84)(34,91,45,78)(35,85,46,79)(50,119,63,106)(51,113,57,107)(52,114,58,108)(53,115,59,109)(54,116,60,110)(55,117,61,111)(56,118,62,112)(64,131,75,120)(65,132,76,121)(66,133,77,122)(67,127,71,123)(68,128,72,124)(69,129,73,125)(70,130,74,126)(134,210,147,214)(135,204,141,215)(136,205,142,216)(137,206,143,217)(138,207,144,211)(139,208,145,212)(140,209,146,213)(148,191,159,197)(149,192,160,198)(150,193,161,199)(151,194,155,200)(152,195,156,201)(153,196,157,202)(154,190,158,203), (8,25)(9,26)(10,27)(11,28)(12,22)(13,23)(14,24)(15,223)(16,224)(17,218)(18,219)(19,220)(20,221)(21,222)(78,102)(79,103)(80,104)(81,105)(82,99)(83,100)(84,101)(85,92)(86,93)(87,94)(88,95)(89,96)(90,97)(91,98)(106,130)(107,131)(108,132)(109,133)(110,127)(111,128)(112,129)(113,120)(114,121)(115,122)(116,123)(117,124)(118,125)(119,126)(134,154)(135,148)(136,149)(137,150)(138,151)(139,152)(140,153)(141,159)(142,160)(143,161)(144,155)(145,156)(146,157)(147,158)(162,182)(163,176)(164,177)(165,178)(166,179)(167,180)(168,181)(169,187)(170,188)(171,189)(172,183)(173,184)(174,185)(175,186)(190,203)(191,197)(192,198)(193,199)(194,200)(195,201)(196,202)(204,215)(205,216)(206,217)(207,211)(208,212)(209,213)(210,214), (1,50,34,74)(2,51,35,75)(3,52,29,76)(4,53,30,77)(5,54,31,71)(6,55,32,72)(7,56,33,73)(8,191,219,215)(9,192,220,216)(10,193,221,217)(11,194,222,211)(12,195,223,212)(13,196,224,213)(14,190,218,214)(15,208,22,201)(16,209,23,202)(17,210,24,203)(18,204,25,197)(19,205,26,198)(20,206,27,199)(21,207,28,200)(36,57,46,64)(37,58,47,65)(38,59,48,66)(39,60,49,67)(40,61,43,68)(41,62,44,69)(42,63,45,70)(78,130,102,106)(79,131,103,107)(80,132,104,108)(81,133,105,109)(82,127,99,110)(83,128,100,111)(84,129,101,112)(85,120,92,113)(86,121,93,114)(87,122,94,115)(88,123,95,116)(89,124,96,117)(90,125,97,118)(91,126,98,119)(134,162,158,186)(135,163,159,187)(136,164,160,188)(137,165,161,189)(138,166,155,183)(139,167,156,184)(140,168,157,185)(141,169,148,176)(142,170,149,177)(143,171,150,178)(144,172,151,179)(145,173,152,180)(146,174,153,181)(147,175,154,182) );
 
G=PermutationGroup([(1,2,3,4,5,6,7),(8,9,10,11,12,13,14),(15,16,17,18,19,20,21),(22,23,24,25,26,27,28),(29,30,31,32,33,34,35),(36,37,38,39,40,41,42),(43,44,45,46,47,48,49),(50,51,52,53,54,55,56),(57,58,59,60,61,62,63),(64,65,66,67,68,69,70),(71,72,73,74,75,76,77),(78,79,80,81,82,83,84),(85,86,87,88,89,90,91),(92,93,94,95,96,97,98),(99,100,101,102,103,104,105),(106,107,108,109,110,111,112),(113,114,115,116,117,118,119),(120,121,122,123,124,125,126),(127,128,129,130,131,132,133),(134,135,136,137,138,139,140),(141,142,143,144,145,146,147),(148,149,150,151,152,153,154),(155,156,157,158,159,160,161),(162,163,164,165,166,167,168),(169,170,171,172,173,174,175),(176,177,178,179,180,181,182),(183,184,185,186,187,188,189),(190,191,192,193,194,195,196),(197,198,199,200,201,202,203),(204,205,206,207,208,209,210),(211,212,213,214,215,216,217),(218,219,220,221,222,223,224)], [(1,34),(2,35),(3,29),(4,30),(5,31),(6,32),(7,33),(8,219),(9,220),(10,221),(11,222),(12,223),(13,224),(14,218),(15,22),(16,23),(17,24),(18,25),(19,26),(20,27),(21,28),(36,46),(37,47),(38,48),(39,49),(40,43),(41,44),(42,45),(50,74),(51,75),(52,76),(53,77),(54,71),(55,72),(56,73),(57,64),(58,65),(59,66),(60,67),(61,68),(62,69),(63,70),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(134,158),(135,159),(136,160),(137,161),(138,155),(139,156),(140,157),(141,148),(142,149),(143,150),(144,151),(145,152),(146,153),(147,154),(162,186),(163,187),(164,188),(165,189),(166,183),(167,184),(168,185),(169,176),(170,177),(171,178),(172,179),(173,180),(174,181),(175,182),(190,214),(191,215),(192,216),(193,217),(194,211),(195,212),(196,213),(197,204),(198,205),(199,206),(200,207),(201,208),(202,209),(203,210)], [(1,45),(2,46),(3,47),(4,48),(5,49),(6,43),(7,44),(8,18),(9,19),(10,20),(11,21),(12,15),(13,16),(14,17),(22,223),(23,224),(24,218),(25,219),(26,220),(27,221),(28,222),(29,37),(30,38),(31,39),(32,40),(33,41),(34,42),(35,36),(50,70),(51,64),(52,65),(53,66),(54,67),(55,68),(56,69),(57,75),(58,76),(59,77),(60,71),(61,72),(62,73),(63,74),(78,98),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,103),(86,104),(87,105),(88,99),(89,100),(90,101),(91,102),(106,126),(107,120),(108,121),(109,122),(110,123),(111,124),(112,125),(113,131),(114,132),(115,133),(116,127),(117,128),(118,129),(119,130),(134,154),(135,148),(136,149),(137,150),(138,151),(139,152),(140,153),(141,159),(142,160),(143,161),(144,155),(145,156),(146,157),(147,158),(162,182),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,210),(191,204),(192,205),(193,206),(194,207),(195,208),(196,209),(197,215),(198,216),(199,217),(200,211),(201,212),(202,213),(203,214)], [(1,158),(2,159),(3,160),(4,161),(5,155),(6,156),(7,157),(8,131),(9,132),(10,133),(11,127),(12,128),(13,129),(14,130),(15,117),(16,118),(17,119),(18,113),(19,114),(20,115),(21,116),(22,124),(23,125),(24,126),(25,120),(26,121),(27,122),(28,123),(29,136),(30,137),(31,138),(32,139),(33,140),(34,134),(35,135),(36,148),(37,149),(38,150),(39,151),(40,152),(41,153),(42,154),(43,145),(44,146),(45,147),(46,141),(47,142),(48,143),(49,144),(50,162),(51,163),(52,164),(53,165),(54,166),(55,167),(56,168),(57,169),(58,170),(59,171),(60,172),(61,173),(62,174),(63,175),(64,176),(65,177),(66,178),(67,179),(68,180),(69,181),(70,182),(71,183),(72,184),(73,185),(74,186),(75,187),(76,188),(77,189),(78,190),(79,191),(80,192),(81,193),(82,194),(83,195),(84,196),(85,197),(86,198),(87,199),(88,200),(89,201),(90,202),(91,203),(92,204),(93,205),(94,206),(95,207),(96,208),(97,209),(98,210),(99,211),(100,212),(101,213),(102,214),(103,215),(104,216),(105,217),(106,218),(107,219),(108,220),(109,221),(110,222),(111,223),(112,224)], [(1,98,42,102),(2,92,36,103),(3,93,37,104),(4,94,38,105),(5,95,39,99),(6,96,40,100),(7,97,41,101),(8,163,25,169),(9,164,26,170),(10,165,27,171),(11,166,28,172),(12,167,22,173),(13,168,23,174),(14,162,24,175),(15,180,223,184),(16,181,224,185),(17,182,218,186),(18,176,219,187),(19,177,220,188),(20,178,221,189),(21,179,222,183),(29,86,47,80),(30,87,48,81),(31,88,49,82),(32,89,43,83),(33,90,44,84),(34,91,45,78),(35,85,46,79),(50,119,63,106),(51,113,57,107),(52,114,58,108),(53,115,59,109),(54,116,60,110),(55,117,61,111),(56,118,62,112),(64,131,75,120),(65,132,76,121),(66,133,77,122),(67,127,71,123),(68,128,72,124),(69,129,73,125),(70,130,74,126),(134,210,147,214),(135,204,141,215),(136,205,142,216),(137,206,143,217),(138,207,144,211),(139,208,145,212),(140,209,146,213),(148,191,159,197),(149,192,160,198),(150,193,161,199),(151,194,155,200),(152,195,156,201),(153,196,157,202),(154,190,158,203)], [(8,25),(9,26),(10,27),(11,28),(12,22),(13,23),(14,24),(15,223),(16,224),(17,218),(18,219),(19,220),(20,221),(21,222),(78,102),(79,103),(80,104),(81,105),(82,99),(83,100),(84,101),(85,92),(86,93),(87,94),(88,95),(89,96),(90,97),(91,98),(106,130),(107,131),(108,132),(109,133),(110,127),(111,128),(112,129),(113,120),(114,121),(115,122),(116,123),(117,124),(118,125),(119,126),(134,154),(135,148),(136,149),(137,150),(138,151),(139,152),(140,153),(141,159),(142,160),(143,161),(144,155),(145,156),(146,157),(147,158),(162,182),(163,176),(164,177),(165,178),(166,179),(167,180),(168,181),(169,187),(170,188),(171,189),(172,183),(173,184),(174,185),(175,186),(190,203),(191,197),(192,198),(193,199),(194,200),(195,201),(196,202),(204,215),(205,216),(206,217),(207,211),(208,212),(209,213),(210,214)], [(1,50,34,74),(2,51,35,75),(3,52,29,76),(4,53,30,77),(5,54,31,71),(6,55,32,72),(7,56,33,73),(8,191,219,215),(9,192,220,216),(10,193,221,217),(11,194,222,211),(12,195,223,212),(13,196,224,213),(14,190,218,214),(15,208,22,201),(16,209,23,202),(17,210,24,203),(18,204,25,197),(19,205,26,198),(20,206,27,199),(21,207,28,200),(36,57,46,64),(37,58,47,65),(38,59,48,66),(39,60,49,67),(40,61,43,68),(41,62,44,69),(42,63,45,70),(78,130,102,106),(79,131,103,107),(80,132,104,108),(81,133,105,109),(82,127,99,110),(83,128,100,111),(84,129,101,112),(85,120,92,113),(86,121,93,114),(87,122,94,115),(88,123,95,116),(89,124,96,117),(90,125,97,118),(91,126,98,119),(134,162,158,186),(135,163,159,187),(136,164,160,188),(137,165,161,189),(138,166,155,183),(139,167,156,184),(140,168,157,185),(141,169,148,176),(142,170,149,177),(143,171,150,178),(144,172,151,179),(145,173,152,180),(146,174,153,181),(147,175,154,182)])
 

Matrix representation ►G ⊆ GL6(𝔽29)

100000
010000
007000
000700
000070
000007
,
100000
010000
0028000
0002800
0000280
0000028
,
2800000
0280000
0028000
0002800
0000280
0000028
,
4250000
11250000
00002716
0000162
0021300
00132700
,
1700000
0170000
000010
000001
001000
000100
,
100000
2280000
001000
000100
0000280
0000028
,
100000
010000
000100
0028000
000001
0000280

G:=sub<GL(6,GF(29))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7,0,0,0,0,0,0,7],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[4,11,0,0,0,0,25,25,0,0,0,0,0,0,0,0,2,13,0,0,0,0,13,27,0,0,27,16,0,0,0,0,16,2,0,0],[17,0,0,0,0,0,0,17,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0],[1,2,0,0,0,0,0,28,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,28],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,28,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,1,0] >;
 

154 conjugacy classes

class 1 2A2B2C2D2E2F4A···4F4G···4O7A···7F14A···14R14S···14AJ28A···28AJ28AK···28CL
order12222224···44···47···714···1414···1428···2828···28
size11114442···24···41···11···14···42···24···4

154 irreducible representations

dim11111111111111111111224444
type+++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C7C14C14C14C14C14C14C14C14C14C4○D4C7×C4○D42+ (1+4)2- (1+4)C7×2+ (1+4)C7×2- (1+4)
kernelC7×C22.36C24C7×C42⋊C2D4×C28Q8×C28C7×C4⋊D4C7×C22⋊Q8C7×C22.D4C7×C4.4D4C7×C42⋊2C2C7×C4⋊Q8C22.36C24C42⋊C2C4×D4C4×Q8C4⋊D4C22⋊Q8C22.D4C4.4D4C42⋊2C2C4⋊Q8C28C4C14C14C2C2
# reps1111132321666661812181264241166

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{36}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.36C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1325);
 
// by ID
 
G=gap.SmallGroup(448,1325);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,2360,4790,1227,3363,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=f^2=1,e^2=c*b=b*c,g^2=b,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,e*d*e^-1=g*d*g^-1=b*d=d*b,f*e*f=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,e*g=g*e,f*g=g*f>;
 
// generators/relations
 

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