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G = C62.114D4order 288 = 25·32

19th non-split extension by C62 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial

Aliases: C12.18D12, C62.114D4, C12.27(C4×S3), (C2×C12).88D6, (C3×C12).50D4, (C3×C6).16Q16, C324Q88C4, C6.23(D6⋊C4), (C3×C6).23SD16, C32(C6.SD16), C6.12(D4.S3), (C6×C12).55C22, C6.11(C3⋊Q16), C329(Q8⋊C4), C4.10(C12⋊S3), C2.2(C327Q16), C2.2(C329SD16), C2.6(C6.11D12), C22.15(C327D4), C4.2(C4×C3⋊S3), C4⋊C4.3(C3⋊S3), (C3×C4⋊C4).19S3, (C3×C12).49(C2×C4), (C32×C4⋊C4).6C2, (C2×C6).90(C3⋊D4), (C2×C324C8).7C2, (C2×C324Q8).9C2, (C3×C6).54(C22⋊C4), (C2×C4).37(C2×C3⋊S3), SmallGroup(288,285)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C62.114D4
C1C3C32C3×C6C62C6×C12C2×C324Q8 — C62.114D4
C32C3×C6C3×C12 — C62.114D4
C1C22C2×C4C4⋊C4

Generators and relations for C62.114D4
 G = < a,b,c,d | a6=b6=1, c4=d2=b3, ab=ba, cac-1=dad-1=a-1, cbc-1=dbd-1=b-1, dcd-1=a3c3 >

Subgroups: 420 in 126 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×3], C22, C6 [×12], C8, C2×C4, C2×C4 [×2], Q8 [×3], C32, Dic3 [×8], C12 [×8], C12 [×4], C2×C6 [×4], C4⋊C4, C2×C8, C2×Q8, C3×C6 [×3], C3⋊C8 [×4], Dic6 [×12], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], Q8⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C3×C12, C62, C2×C3⋊C8 [×4], C3×C4⋊C4 [×4], C2×Dic6 [×4], C324C8, C324Q8 [×2], C324Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C6.SD16 [×4], C2×C324C8, C32×C4⋊C4, C2×C324Q8, C62.114D4
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4 [×2], D6 [×4], C22⋊C4, SD16, Q16, C3⋊S3, C4×S3 [×4], D12 [×4], C3⋊D4 [×4], Q8⋊C4, C2×C3⋊S3, D6⋊C4 [×4], D4.S3 [×4], C3⋊Q16 [×4], C4×C3⋊S3, C12⋊S3, C327D4, C6.SD16 [×4], C6.11D12, C329SD16, C327Q16, C62.114D4

Smallest permutation representation of C62.114D4
Regular action on 288 points
Generators in S288
(1 56 285 131 187 65)(2 66 188 132 286 49)(3 50 287 133 189 67)(4 68 190 134 288 51)(5 52 281 135 191 69)(6 70 192 136 282 53)(7 54 283 129 185 71)(8 72 186 130 284 55)(9 96 139 174 222 273)(10 274 223 175 140 89)(11 90 141 176 224 275)(12 276 217 169 142 91)(13 92 143 170 218 277)(14 278 219 171 144 93)(15 94 137 172 220 279)(16 280 221 173 138 95)(17 41 128 106 207 34)(18 35 208 107 121 42)(19 43 122 108 201 36)(20 37 202 109 123 44)(21 45 124 110 203 38)(22 39 204 111 125 46)(23 47 126 112 205 40)(24 33 206 105 127 48)(25 117 168 147 235 254)(26 255 236 148 161 118)(27 119 162 149 237 256)(28 249 238 150 163 120)(29 113 164 151 239 250)(30 251 240 152 165 114)(31 115 166 145 233 252)(32 253 234 146 167 116)(57 268 231 87 75 183)(58 184 76 88 232 269)(59 270 225 81 77 177)(60 178 78 82 226 271)(61 272 227 83 79 179)(62 180 80 84 228 265)(63 266 229 85 73 181)(64 182 74 86 230 267)(97 195 259 245 213 155)(98 156 214 246 260 196)(99 197 261 247 215 157)(100 158 216 248 262 198)(101 199 263 241 209 159)(102 160 210 242 264 200)(103 193 257 243 211 153)(104 154 212 244 258 194)
(1 63 172 5 59 176)(2 169 60 6 173 64)(3 57 174 7 61 170)(4 171 62 8 175 58)(9 129 83 13 133 87)(10 88 134 14 84 130)(11 131 85 15 135 81)(12 82 136 16 86 132)(17 27 213 21 31 209)(18 210 32 22 214 28)(19 29 215 23 25 211)(20 212 26 24 216 30)(33 248 251 37 244 255)(34 256 245 38 252 241)(35 242 253 39 246 249)(36 250 247 40 254 243)(41 119 155 45 115 159)(42 160 116 46 156 120)(43 113 157 47 117 153)(44 154 118 48 158 114)(49 217 271 53 221 267)(50 268 222 54 272 218)(51 219 265 55 223 269)(52 270 224 56 266 220)(65 181 137 69 177 141)(66 142 178 70 138 182)(67 183 139 71 179 143)(68 144 180 72 140 184)(73 94 191 77 90 187)(74 188 91 78 192 95)(75 96 185 79 92 189)(76 190 93 80 186 89)(97 124 166 101 128 162)(98 163 121 102 167 125)(99 126 168 103 122 164)(100 165 123 104 161 127)(105 198 152 109 194 148)(106 149 195 110 145 199)(107 200 146 111 196 150)(108 151 197 112 147 193)(201 239 261 205 235 257)(202 258 236 206 262 240)(203 233 263 207 237 259)(204 260 238 208 264 234)(225 275 285 229 279 281)(226 282 280 230 286 276)(227 277 287 231 273 283)(228 284 274 232 288 278)
(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)(225 226 227 228 229 230 231 232)(233 234 235 236 237 238 239 240)(241 242 243 244 245 246 247 248)(249 250 251 252 253 254 255 256)(257 258 259 260 261 262 263 264)(265 266 267 268 269 270 271 272)(273 274 275 276 277 278 279 280)(281 282 283 284 285 286 287 288)
(1 240 5 236)(2 117 6 113)(3 238 7 234)(4 115 8 119)(9 46 13 42)(10 207 14 203)(11 44 15 48)(12 205 16 201)(17 144 21 140)(18 273 22 277)(19 142 23 138)(20 279 24 275)(25 70 29 66)(26 285 30 281)(27 68 31 72)(28 283 32 287)(33 224 37 220)(34 93 38 89)(35 222 39 218)(36 91 40 95)(41 171 45 175)(43 169 47 173)(49 168 53 164)(50 249 54 253)(51 166 55 162)(52 255 56 251)(57 260 61 264)(58 159 62 155)(59 258 63 262)(60 157 64 153)(65 152 69 148)(67 150 71 146)(73 100 77 104)(74 243 78 247)(75 98 79 102)(76 241 80 245)(81 154 85 158)(82 261 86 257)(83 160 87 156)(84 259 88 263)(90 123 94 127)(92 121 96 125)(97 269 101 265)(99 267 103 271)(105 141 109 137)(106 278 110 274)(107 139 111 143)(108 276 112 280)(114 135 118 131)(116 133 120 129)(122 217 126 221)(124 223 128 219)(130 237 134 233)(132 235 136 239)(145 284 149 288)(147 282 151 286)(161 187 165 191)(163 185 167 189)(170 208 174 204)(172 206 176 202)(177 194 181 198)(178 215 182 211)(179 200 183 196)(180 213 184 209)(186 256 190 252)(188 254 192 250)(193 226 197 230)(195 232 199 228)(210 231 214 227)(212 229 216 225)(242 268 246 272)(244 266 248 270)

G:=sub<Sym(288)| (1,56,285,131,187,65)(2,66,188,132,286,49)(3,50,287,133,189,67)(4,68,190,134,288,51)(5,52,281,135,191,69)(6,70,192,136,282,53)(7,54,283,129,185,71)(8,72,186,130,284,55)(9,96,139,174,222,273)(10,274,223,175,140,89)(11,90,141,176,224,275)(12,276,217,169,142,91)(13,92,143,170,218,277)(14,278,219,171,144,93)(15,94,137,172,220,279)(16,280,221,173,138,95)(17,41,128,106,207,34)(18,35,208,107,121,42)(19,43,122,108,201,36)(20,37,202,109,123,44)(21,45,124,110,203,38)(22,39,204,111,125,46)(23,47,126,112,205,40)(24,33,206,105,127,48)(25,117,168,147,235,254)(26,255,236,148,161,118)(27,119,162,149,237,256)(28,249,238,150,163,120)(29,113,164,151,239,250)(30,251,240,152,165,114)(31,115,166,145,233,252)(32,253,234,146,167,116)(57,268,231,87,75,183)(58,184,76,88,232,269)(59,270,225,81,77,177)(60,178,78,82,226,271)(61,272,227,83,79,179)(62,180,80,84,228,265)(63,266,229,85,73,181)(64,182,74,86,230,267)(97,195,259,245,213,155)(98,156,214,246,260,196)(99,197,261,247,215,157)(100,158,216,248,262,198)(101,199,263,241,209,159)(102,160,210,242,264,200)(103,193,257,243,211,153)(104,154,212,244,258,194), (1,63,172,5,59,176)(2,169,60,6,173,64)(3,57,174,7,61,170)(4,171,62,8,175,58)(9,129,83,13,133,87)(10,88,134,14,84,130)(11,131,85,15,135,81)(12,82,136,16,86,132)(17,27,213,21,31,209)(18,210,32,22,214,28)(19,29,215,23,25,211)(20,212,26,24,216,30)(33,248,251,37,244,255)(34,256,245,38,252,241)(35,242,253,39,246,249)(36,250,247,40,254,243)(41,119,155,45,115,159)(42,160,116,46,156,120)(43,113,157,47,117,153)(44,154,118,48,158,114)(49,217,271,53,221,267)(50,268,222,54,272,218)(51,219,265,55,223,269)(52,270,224,56,266,220)(65,181,137,69,177,141)(66,142,178,70,138,182)(67,183,139,71,179,143)(68,144,180,72,140,184)(73,94,191,77,90,187)(74,188,91,78,192,95)(75,96,185,79,92,189)(76,190,93,80,186,89)(97,124,166,101,128,162)(98,163,121,102,167,125)(99,126,168,103,122,164)(100,165,123,104,161,127)(105,198,152,109,194,148)(106,149,195,110,145,199)(107,200,146,111,196,150)(108,151,197,112,147,193)(201,239,261,205,235,257)(202,258,236,206,262,240)(203,233,263,207,237,259)(204,260,238,208,264,234)(225,275,285,229,279,281)(226,282,280,230,286,276)(227,277,287,231,273,283)(228,284,274,232,288,278), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,240,5,236)(2,117,6,113)(3,238,7,234)(4,115,8,119)(9,46,13,42)(10,207,14,203)(11,44,15,48)(12,205,16,201)(17,144,21,140)(18,273,22,277)(19,142,23,138)(20,279,24,275)(25,70,29,66)(26,285,30,281)(27,68,31,72)(28,283,32,287)(33,224,37,220)(34,93,38,89)(35,222,39,218)(36,91,40,95)(41,171,45,175)(43,169,47,173)(49,168,53,164)(50,249,54,253)(51,166,55,162)(52,255,56,251)(57,260,61,264)(58,159,62,155)(59,258,63,262)(60,157,64,153)(65,152,69,148)(67,150,71,146)(73,100,77,104)(74,243,78,247)(75,98,79,102)(76,241,80,245)(81,154,85,158)(82,261,86,257)(83,160,87,156)(84,259,88,263)(90,123,94,127)(92,121,96,125)(97,269,101,265)(99,267,103,271)(105,141,109,137)(106,278,110,274)(107,139,111,143)(108,276,112,280)(114,135,118,131)(116,133,120,129)(122,217,126,221)(124,223,128,219)(130,237,134,233)(132,235,136,239)(145,284,149,288)(147,282,151,286)(161,187,165,191)(163,185,167,189)(170,208,174,204)(172,206,176,202)(177,194,181,198)(178,215,182,211)(179,200,183,196)(180,213,184,209)(186,256,190,252)(188,254,192,250)(193,226,197,230)(195,232,199,228)(210,231,214,227)(212,229,216,225)(242,268,246,272)(244,266,248,270)>;

G:=Group( (1,56,285,131,187,65)(2,66,188,132,286,49)(3,50,287,133,189,67)(4,68,190,134,288,51)(5,52,281,135,191,69)(6,70,192,136,282,53)(7,54,283,129,185,71)(8,72,186,130,284,55)(9,96,139,174,222,273)(10,274,223,175,140,89)(11,90,141,176,224,275)(12,276,217,169,142,91)(13,92,143,170,218,277)(14,278,219,171,144,93)(15,94,137,172,220,279)(16,280,221,173,138,95)(17,41,128,106,207,34)(18,35,208,107,121,42)(19,43,122,108,201,36)(20,37,202,109,123,44)(21,45,124,110,203,38)(22,39,204,111,125,46)(23,47,126,112,205,40)(24,33,206,105,127,48)(25,117,168,147,235,254)(26,255,236,148,161,118)(27,119,162,149,237,256)(28,249,238,150,163,120)(29,113,164,151,239,250)(30,251,240,152,165,114)(31,115,166,145,233,252)(32,253,234,146,167,116)(57,268,231,87,75,183)(58,184,76,88,232,269)(59,270,225,81,77,177)(60,178,78,82,226,271)(61,272,227,83,79,179)(62,180,80,84,228,265)(63,266,229,85,73,181)(64,182,74,86,230,267)(97,195,259,245,213,155)(98,156,214,246,260,196)(99,197,261,247,215,157)(100,158,216,248,262,198)(101,199,263,241,209,159)(102,160,210,242,264,200)(103,193,257,243,211,153)(104,154,212,244,258,194), (1,63,172,5,59,176)(2,169,60,6,173,64)(3,57,174,7,61,170)(4,171,62,8,175,58)(9,129,83,13,133,87)(10,88,134,14,84,130)(11,131,85,15,135,81)(12,82,136,16,86,132)(17,27,213,21,31,209)(18,210,32,22,214,28)(19,29,215,23,25,211)(20,212,26,24,216,30)(33,248,251,37,244,255)(34,256,245,38,252,241)(35,242,253,39,246,249)(36,250,247,40,254,243)(41,119,155,45,115,159)(42,160,116,46,156,120)(43,113,157,47,117,153)(44,154,118,48,158,114)(49,217,271,53,221,267)(50,268,222,54,272,218)(51,219,265,55,223,269)(52,270,224,56,266,220)(65,181,137,69,177,141)(66,142,178,70,138,182)(67,183,139,71,179,143)(68,144,180,72,140,184)(73,94,191,77,90,187)(74,188,91,78,192,95)(75,96,185,79,92,189)(76,190,93,80,186,89)(97,124,166,101,128,162)(98,163,121,102,167,125)(99,126,168,103,122,164)(100,165,123,104,161,127)(105,198,152,109,194,148)(106,149,195,110,145,199)(107,200,146,111,196,150)(108,151,197,112,147,193)(201,239,261,205,235,257)(202,258,236,206,262,240)(203,233,263,207,237,259)(204,260,238,208,264,234)(225,275,285,229,279,281)(226,282,280,230,286,276)(227,277,287,231,273,283)(228,284,274,232,288,278), (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)(225,226,227,228,229,230,231,232)(233,234,235,236,237,238,239,240)(241,242,243,244,245,246,247,248)(249,250,251,252,253,254,255,256)(257,258,259,260,261,262,263,264)(265,266,267,268,269,270,271,272)(273,274,275,276,277,278,279,280)(281,282,283,284,285,286,287,288), (1,240,5,236)(2,117,6,113)(3,238,7,234)(4,115,8,119)(9,46,13,42)(10,207,14,203)(11,44,15,48)(12,205,16,201)(17,144,21,140)(18,273,22,277)(19,142,23,138)(20,279,24,275)(25,70,29,66)(26,285,30,281)(27,68,31,72)(28,283,32,287)(33,224,37,220)(34,93,38,89)(35,222,39,218)(36,91,40,95)(41,171,45,175)(43,169,47,173)(49,168,53,164)(50,249,54,253)(51,166,55,162)(52,255,56,251)(57,260,61,264)(58,159,62,155)(59,258,63,262)(60,157,64,153)(65,152,69,148)(67,150,71,146)(73,100,77,104)(74,243,78,247)(75,98,79,102)(76,241,80,245)(81,154,85,158)(82,261,86,257)(83,160,87,156)(84,259,88,263)(90,123,94,127)(92,121,96,125)(97,269,101,265)(99,267,103,271)(105,141,109,137)(106,278,110,274)(107,139,111,143)(108,276,112,280)(114,135,118,131)(116,133,120,129)(122,217,126,221)(124,223,128,219)(130,237,134,233)(132,235,136,239)(145,284,149,288)(147,282,151,286)(161,187,165,191)(163,185,167,189)(170,208,174,204)(172,206,176,202)(177,194,181,198)(178,215,182,211)(179,200,183,196)(180,213,184,209)(186,256,190,252)(188,254,192,250)(193,226,197,230)(195,232,199,228)(210,231,214,227)(212,229,216,225)(242,268,246,272)(244,266,248,270) );

G=PermutationGroup([(1,56,285,131,187,65),(2,66,188,132,286,49),(3,50,287,133,189,67),(4,68,190,134,288,51),(5,52,281,135,191,69),(6,70,192,136,282,53),(7,54,283,129,185,71),(8,72,186,130,284,55),(9,96,139,174,222,273),(10,274,223,175,140,89),(11,90,141,176,224,275),(12,276,217,169,142,91),(13,92,143,170,218,277),(14,278,219,171,144,93),(15,94,137,172,220,279),(16,280,221,173,138,95),(17,41,128,106,207,34),(18,35,208,107,121,42),(19,43,122,108,201,36),(20,37,202,109,123,44),(21,45,124,110,203,38),(22,39,204,111,125,46),(23,47,126,112,205,40),(24,33,206,105,127,48),(25,117,168,147,235,254),(26,255,236,148,161,118),(27,119,162,149,237,256),(28,249,238,150,163,120),(29,113,164,151,239,250),(30,251,240,152,165,114),(31,115,166,145,233,252),(32,253,234,146,167,116),(57,268,231,87,75,183),(58,184,76,88,232,269),(59,270,225,81,77,177),(60,178,78,82,226,271),(61,272,227,83,79,179),(62,180,80,84,228,265),(63,266,229,85,73,181),(64,182,74,86,230,267),(97,195,259,245,213,155),(98,156,214,246,260,196),(99,197,261,247,215,157),(100,158,216,248,262,198),(101,199,263,241,209,159),(102,160,210,242,264,200),(103,193,257,243,211,153),(104,154,212,244,258,194)], [(1,63,172,5,59,176),(2,169,60,6,173,64),(3,57,174,7,61,170),(4,171,62,8,175,58),(9,129,83,13,133,87),(10,88,134,14,84,130),(11,131,85,15,135,81),(12,82,136,16,86,132),(17,27,213,21,31,209),(18,210,32,22,214,28),(19,29,215,23,25,211),(20,212,26,24,216,30),(33,248,251,37,244,255),(34,256,245,38,252,241),(35,242,253,39,246,249),(36,250,247,40,254,243),(41,119,155,45,115,159),(42,160,116,46,156,120),(43,113,157,47,117,153),(44,154,118,48,158,114),(49,217,271,53,221,267),(50,268,222,54,272,218),(51,219,265,55,223,269),(52,270,224,56,266,220),(65,181,137,69,177,141),(66,142,178,70,138,182),(67,183,139,71,179,143),(68,144,180,72,140,184),(73,94,191,77,90,187),(74,188,91,78,192,95),(75,96,185,79,92,189),(76,190,93,80,186,89),(97,124,166,101,128,162),(98,163,121,102,167,125),(99,126,168,103,122,164),(100,165,123,104,161,127),(105,198,152,109,194,148),(106,149,195,110,145,199),(107,200,146,111,196,150),(108,151,197,112,147,193),(201,239,261,205,235,257),(202,258,236,206,262,240),(203,233,263,207,237,259),(204,260,238,208,264,234),(225,275,285,229,279,281),(226,282,280,230,286,276),(227,277,287,231,273,283),(228,284,274,232,288,278)], [(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),(225,226,227,228,229,230,231,232),(233,234,235,236,237,238,239,240),(241,242,243,244,245,246,247,248),(249,250,251,252,253,254,255,256),(257,258,259,260,261,262,263,264),(265,266,267,268,269,270,271,272),(273,274,275,276,277,278,279,280),(281,282,283,284,285,286,287,288)], [(1,240,5,236),(2,117,6,113),(3,238,7,234),(4,115,8,119),(9,46,13,42),(10,207,14,203),(11,44,15,48),(12,205,16,201),(17,144,21,140),(18,273,22,277),(19,142,23,138),(20,279,24,275),(25,70,29,66),(26,285,30,281),(27,68,31,72),(28,283,32,287),(33,224,37,220),(34,93,38,89),(35,222,39,218),(36,91,40,95),(41,171,45,175),(43,169,47,173),(49,168,53,164),(50,249,54,253),(51,166,55,162),(52,255,56,251),(57,260,61,264),(58,159,62,155),(59,258,63,262),(60,157,64,153),(65,152,69,148),(67,150,71,146),(73,100,77,104),(74,243,78,247),(75,98,79,102),(76,241,80,245),(81,154,85,158),(82,261,86,257),(83,160,87,156),(84,259,88,263),(90,123,94,127),(92,121,96,125),(97,269,101,265),(99,267,103,271),(105,141,109,137),(106,278,110,274),(107,139,111,143),(108,276,112,280),(114,135,118,131),(116,133,120,129),(122,217,126,221),(124,223,128,219),(130,237,134,233),(132,235,136,239),(145,284,149,288),(147,282,151,286),(161,187,165,191),(163,185,167,189),(170,208,174,204),(172,206,176,202),(177,194,181,198),(178,215,182,211),(179,200,183,196),(180,213,184,209),(186,256,190,252),(188,254,192,250),(193,226,197,230),(195,232,199,228),(210,231,214,227),(212,229,216,225),(242,268,246,272),(244,266,248,270)])

54 conjugacy classes

class 1 2A2B2C3A3B3C3D4A4B4C4D4E4F6A···6L8A8B8C8D12A···12X
order122233334444446···6888812···12
size11112222224436362···2181818184···4

54 irreducible representations

dim1111122222222244
type++++++++-+--
imageC1C2C2C2C4S3D4D4D6SD16Q16C4×S3D12C3⋊D4D4.S3C3⋊Q16
kernelC62.114D4C2×C324C8C32×C4⋊C4C2×C324Q8C324Q8C3×C4⋊C4C3×C12C62C2×C12C3×C6C3×C6C12C12C2×C6C6C6
# reps1111441142288844

Matrix representation of C62.114D4 in GL6(𝔽73)

1720000
100000
001000
000100
000011
0000720
,
0720000
1720000
0072000
0007200
00007272
000010
,
61220000
10120000
0067600
00676700
00003748
00001136
,
32630000
22410000
00656400
0064800
00005055
0000523

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,1,0],[0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,72,0],[61,10,0,0,0,0,22,12,0,0,0,0,0,0,67,67,0,0,0,0,6,67,0,0,0,0,0,0,37,11,0,0,0,0,48,36],[32,22,0,0,0,0,63,41,0,0,0,0,0,0,65,64,0,0,0,0,64,8,0,0,0,0,0,0,50,5,0,0,0,0,55,23] >;

C62.114D4 in GAP, Magma, Sage, TeX

C_6^2._{114}D_4
% in TeX

G:=Group("C6^2.114D4");
// GroupNames label

G:=SmallGroup(288,285);
// by ID

G=gap.SmallGroup(288,285);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,675,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^6=1,c^4=d^2=b^3,a*b=b*a,c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=a^3*c^3>;
// generators/relations

׿
×
𝔽