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G = C7×C22.50C24  order 448 = 26·7

Direct product of C7 and C22.50C24

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

Aliases: C7×C22.50C24, C14.1212- (1+4), C4⋊Q8⋊17C14, (Q8×C28)⋊36C2, (C4×Q8)⋊16C14, (D4×C28).28C2, (C4×D4).13C14, C22⋊Q8⋊18C14, C42⋊2C2⋊7C14, C42.50(C2×C14), C4.4D4.8C14, C42⋊C2⋊18C14, C28.347(C4○D4), (C2×C28).965C23, (C4×C28).291C22, (C2×C14).376C24, (D4×C14).325C22, C22.50(C23×C14), C23.21(C22×C14), (Q8×C14).277C22, C2.13(C7×2- (1+4)), (C22×C28).461C22, (C22×C14).104C23, (C7×C4⋊Q8)⋊38C2, C4.39(C7×C4○D4), C4⋊C4.76(C2×C14), C2.29(C14×C4○D4), (C7×C22⋊Q8)⋊45C2, (C2×D4).71(C2×C14), C14.248(C2×C4○D4), C22⋊C4.6(C2×C14), (C2×Q8).65(C2×C14), (C7×C42⋊2C2)⋊18C2, (C7×C42⋊C2)⋊39C2, (C7×C4⋊C4).401C22, (C2×C4).38(C22×C14), (C22×C4).72(C2×C14), (C7×C4.4D4).17C2, (C7×C22⋊C4).90C22, SmallGroup(448,1339)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.50C24
Chief series
C1 — C2 — C22 — C2×C14 — C22×C14 — C7×C22⋊C4 — C7×C42⋊2C2 — C7×C22.50C24
Lower central
C1 — C22 — C7×C22.50C24
Upper central
C1 — C2×C14 — C7×C22.50C24

Subgroups: 282 in 212 conjugacy classes, 150 normal (34 characteristic)
C1, C2 [×3], C2 [×2], C4 [×4], C4 [×11], C22, C22 [×6], C7, C2×C4 [×3], C2×C4 [×10], C2×C4 [×4], D4 [×2], Q8 [×6], C23 [×2], C14 [×3], C14 [×2], C42, C42 [×6], C22⋊C4 [×10], C4⋊C4 [×2], C4⋊C4 [×10], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C28 [×4], C28 [×11], C2×C14, C2×C14 [×6], C42⋊C2 [×2], C4×D4, C4×Q8, C4×Q8 [×2], C22⋊Q8 [×2], C4.4D4 [×2], C42⋊2C2 [×4], C4⋊Q8, C2×C28 [×3], C2×C28 [×10], C2×C28 [×4], C7×D4 [×2], C7×Q8 [×6], C22×C14 [×2], C22.50C24, C4×C28, C4×C28 [×6], C7×C22⋊C4 [×10], C7×C4⋊C4 [×2], C7×C4⋊C4 [×10], C22×C28 [×2], D4×C14, Q8×C14, Q8×C14 [×2], C7×C42⋊C2 [×2], D4×C28, Q8×C28, Q8×C28 [×2], C7×C22⋊Q8 [×2], C7×C4.4D4 [×2], C7×C42⋊2C2 [×4], C7×C4⋊Q8, C7×C22.50C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C4○D4 [×4], C24, C2×C14 [×35], C2×C4○D4 [×2], 2- (1+4), C22×C14 [×15], C22.50C24, C7×C4○D4 [×4], C23×C14, C14×C4○D4 [×2], C7×2- (1+4), C7×C22.50C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=d2=1, e2=cb=bc, f2=g2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, ede-1=bd=db, geg-1=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, dg=gd, ef=fe, 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)(8 222)(9 223)(10 224)(11 218)(12 219)(13 220)(14 221)(15 22)(16 23)(17 24)(18 25)(19 26)(20 27)(21 28)(36 43)(37 44)(38 45)(39 46)(40 47)(41 48)(42 49)(50 74)(51 75)(52 76)(53 77)(54 71)(55 72)(56 73)(57 68)(58 69)(59 70)(60 64)(61 65)(62 66)(63 67)(78 102)(79 103)(80 104)(81 105)(82 99)(83 100)(84 101)(85 96)(86 97)(87 98)(88 92)(89 93)(90 94)(91 95)(106 130)(107 131)(108 132)(109 133)(110 127)(111 128)(112 129)(113 124)(114 125)(115 126)(116 120)(117 121)(118 122)(119 123)(134 158)(135 159)(136 160)(137 161)(138 155)(139 156)(140 157)(141 152)(142 153)(143 154)(144 148)(145 149)(146 150)(147 151)(162 186)(163 187)(164 188)(165 189)(166 183)(167 184)(168 185)(169 180)(170 181)(171 182)(172 176)(173 177)(174 178)(175 179)(190 214)(191 215)(192 216)(193 217)(194 211)(195 212)(196 213)(197 208)(198 209)(199 210)(200 204)(201 205)(202 206)(203 207)

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

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

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

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

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

Matrix representation ►G ⊆ GL4(𝔽29) generated by

16000
01600
00160
00016
,
1000
0100
00280
00028
,
28000
02800
0010
0001
,
1000
02800
0010
001228
,
12000
01200
00172
00012
,
0100
1000
00120
00012
,
28000
02800
00170
00112
G:=sub<GL(4,GF(29))| [16,0,0,0,0,16,0,0,0,0,16,0,0,0,0,16],[1,0,0,0,0,1,0,0,0,0,28,0,0,0,0,28],[28,0,0,0,0,28,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,28,0,0,0,0,1,12,0,0,0,28],[12,0,0,0,0,12,0,0,0,0,17,0,0,0,2,12],[0,1,0,0,1,0,0,0,0,0,12,0,0,0,0,12],[28,0,0,0,0,28,0,0,0,0,17,1,0,0,0,12] >;
 

175 conjugacy classes

class 1 2A2B2C2D2E4A···4L4M···4S7A···7F14A···14R14S···14AD28A···28BT28BU···28DJ
order1222224···44···47···714···1414···1428···2828···28
size1111442···24···41···11···14···42···24···4

175 irreducible representations

dim11111111111111112244
type++++++++-
imageC1C2C2C2C2C2C2C2C7C14C14C14C14C14C14C14C4○D4C7×C4○D42- (1+4)C7×2- (1+4)
kernelC7×C22.50C24C7×C42⋊C2D4×C28Q8×C28C7×C22⋊Q8C7×C4.4D4C7×C42⋊2C2C7×C4⋊Q8C22.50C24C42⋊C2C4×D4C4×Q8C22⋊Q8C4.4D4C42⋊2C2C4⋊Q8C28C4C14C2
# reps12132241612618121224684816

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{50}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.50C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1339);
 
// by ID
 
G=gap.SmallGroup(448,1339);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,1597,2360,4790,604,1690,416]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=d^2=1,e^2=c*b=b*c,f^2=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=b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,f*d*f^-1=c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,d*g=g*d,e*f=f*e,f*g=g*f>;
 
// generators/relations
 

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×
⋊
ℤ
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