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G = C7×C22.57C24  order 448 = 26·7

Direct product of C7 and C22.57C24

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

Aliases: C7×C22.57C24, C14.1242- (1+4), C14.1722+ (1+4), C4⋊Q8⋊18C14, C22⋊Q8⋊20C14, C42⋊2C2⋊9C14, C42.55(C2×C14), C42.C2⋊12C14, C4.4D4.9C14, (C2×C28).684C23, (C4×C28).296C22, (C2×C14).383C24, (D4×C14).225C22, C23.26(C22×C14), C22.57(C23×C14), (Q8×C14).188C22, C22.D4.3C14, C2.24(C7×2+ (1+4)), C2.16(C7×2- (1+4)), (C22×C14).109C23, (C22×C28).463C22, (C7×C4⋊Q8)⋊39C2, C4⋊C4.35(C2×C14), (C7×C22⋊Q8)⋊47C2, (C2×D4).38(C2×C14), C22⋊C4.8(C2×C14), (C2×Q8).31(C2×C14), (C7×C42.C2)⋊29C2, (C7×C42⋊2C2)⋊20C2, (C7×C4⋊C4).252C22, (C22×C4).74(C2×C14), (C2×C4).43(C22×C14), (C7×C4.4D4).18C2, (C7×C22⋊C4).93C22, (C7×C22.D4).6C2, SmallGroup(448,1346)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C7×C22.57C24
Chief series
C1 — C2 — C22 — C2×C14 — C2×C28 — Q8×C14 — C7×C4⋊Q8 — C7×C22.57C24
Lower central
C1 — C22 — C7×C22.57C24
Upper central
C1 — C2×C14 — C7×C22.57C24

Subgroups: 282 in 196 conjugacy classes, 142 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×13], C22, C22 [×6], C7, C2×C4, C2×C4 [×12], C2×C4 [×2], D4, Q8 [×3], C23 [×2], C14, C14 [×2], C14 [×2], C42, C42 [×2], C22⋊C4 [×10], C4⋊C4 [×16], C22×C4 [×2], C2×D4, C2×Q8, C2×Q8 [×2], C28 [×13], C2×C14, C2×C14 [×6], C22⋊Q8 [×4], C22.D4 [×2], C4.4D4, C42.C2 [×2], C42⋊2C2 [×4], C4⋊Q8 [×2], C2×C28, C2×C28 [×12], C2×C28 [×2], C7×D4, C7×Q8 [×3], C22×C14 [×2], C22.57C24, C4×C28, C4×C28 [×2], C7×C22⋊C4 [×10], C7×C4⋊C4 [×16], C22×C28 [×2], D4×C14, Q8×C14, Q8×C14 [×2], C7×C22⋊Q8 [×4], C7×C22.D4 [×2], C7×C4.4D4, C7×C42.C2 [×2], C7×C42⋊2C2 [×4], C7×C4⋊Q8 [×2], C7×C22.57C24

Quotients:
C1, C2 [×15], C22 [×35], C7, C23 [×15], C14 [×15], C24, C2×C14 [×35], 2+ (1+4), 2- (1+4) [×2], C22×C14 [×15], C22.57C24, C23×C14, C7×2+ (1+4), C7×2- (1+4) [×2], C7×C22.57C24

Generators and relations
 G = < a,b,c,d,e,f,g | a7=b2=c2=g2=1, d2=e2=f2=b, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, ede-1=bd=db, geg=be=eb, bf=fb, bg=gb, fdf-1=cd=dc, ce=ec, cf=fc, cg=gc, gdg=bcd, fef-1=bce, 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 39)(2 40)(3 41)(4 42)(5 36)(6 37)(7 38)(8 23)(9 24)(10 25)(11 26)(12 27)(13 28)(14 22)(15 221)(16 222)(17 223)(18 224)(19 218)(20 219)(21 220)(29 48)(30 49)(31 43)(32 44)(33 45)(34 46)(35 47)(50 66)(51 67)(52 68)(53 69)(54 70)(55 64)(56 65)(57 76)(58 77)(59 71)(60 72)(61 73)(62 74)(63 75)(78 94)(79 95)(80 96)(81 97)(82 98)(83 92)(84 93)(85 104)(86 105)(87 99)(88 100)(89 101)(90 102)(91 103)(106 122)(107 123)(108 124)(109 125)(110 126)(111 120)(112 121)(113 132)(114 133)(115 127)(116 128)(117 129)(118 130)(119 131)(134 150)(135 151)(136 152)(137 153)(138 154)(139 148)(140 149)(141 160)(142 161)(143 155)(144 156)(145 157)(146 158)(147 159)(162 178)(163 179)(164 180)(165 181)(166 182)(167 176)(168 177)(169 188)(170 189)(171 183)(172 184)(173 185)(174 186)(175 187)(190 206)(191 207)(192 208)(193 209)(194 210)(195 204)(196 205)(197 216)(198 217)(199 211)(200 212)(201 213)(202 214)(203 215)

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

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

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

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

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

Matrix representation ►G ⊆ GL8(𝔽29)

200000000
020000000
002000000
000200000
000016000
000001600
000000160
000000016
,
10000000
01000000
00100000
00010000
000028000
000002800
000000280
000000028
,
280000000
028000000
002800000
000280000
00001000
00000100
00000010
00000001
,
71911100000
20209270000
27022100000
027990000
000099911
00001111277
000018900
0000110189
,
11210190000
271825190000
001120000
0027180000
00000010
0000112827
000028000
000011028
,
015280000
102850000
000280000
002800000
00000100
000028000
0000282812
0000012828
,
107190000
0120200000
002800000
000280000
00001000
00000100
000000280
000011028

G:=sub<GL(8,GF(29))| [20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,20,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16,0,0,0,0,0,0,0,0,16],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28],[28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[7,20,27,0,0,0,0,0,19,20,0,27,0,0,0,0,11,9,22,9,0,0,0,0,10,27,10,9,0,0,0,0,0,0,0,0,9,11,18,11,0,0,0,0,9,11,9,0,0,0,0,0,9,27,0,18,0,0,0,0,11,7,0,9],[11,27,0,0,0,0,0,0,2,18,0,0,0,0,0,0,10,25,11,27,0,0,0,0,19,19,2,18,0,0,0,0,0,0,0,0,0,1,28,1,0,0,0,0,0,1,0,1,0,0,0,0,1,28,0,0,0,0,0,0,0,27,0,28],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,5,28,0,28,0,0,0,0,28,5,28,0,0,0,0,0,0,0,0,0,0,28,28,0,0,0,0,0,1,0,28,1,0,0,0,0,0,0,1,28,0,0,0,0,0,0,2,28],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,7,20,28,0,0,0,0,0,19,20,0,28,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,28,0,0,0,0,0,0,0,0,28] >;
 

133 conjugacy classes

class 1 2A2B2C2D2E4A···4M7A···7F14A···14R14S···14AD28A···28BZ
order1222224···47···714···1414···1428···28
size1111444···41···11···14···44···4

133 irreducible representations

dim111111111111114444
type++++++++-
imageC1C2C2C2C2C2C2C7C14C14C14C14C14C142+ (1+4)2- (1+4)C7×2+ (1+4)C7×2- (1+4)
kernelC7×C22.57C24C7×C22⋊Q8C7×C22.D4C7×C4.4D4C7×C42.C2C7×C42⋊2C2C7×C4⋊Q8C22.57C24C22⋊Q8C22.D4C4.4D4C42.C2C42⋊2C2C4⋊Q8C14C14C2C2
# reps142124262412612241212612

In GAP, Magma, Sage, TeX 

C_7\times C_2^2._{57}C_2^4
 
% in TeX
 
G:=Group("C7xC2^2.57C2^4");
 
// GroupNames label
 
G:=SmallGroup(448,1346);
 
// by ID
 
G=gap.SmallGroup(448,1346);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-7,-2,-2,784,1597,792,4790,3579,604,9635,1690]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^7=b^2=c^2=g^2=1,d^2=e^2=f^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,b*c=c*b,e*d*e^-1=b*d=d*b,g*e*g=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,g*d*g=b*c*d,f*e*f^-1=b*c*e,f*g=g*f>;
 
// generators/relations
 

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