Copied to
clipboard

## G = C12.30Dic6order 288 = 25·32

### 12nd non-split extension by C12 of Dic6 acting via Dic6/C12=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C12.30Dic6
 Chief series C1 — C3 — C32 — C3×C6 — C3×C12 — C6×C12 — C4×C3⋊Dic3 — C12.30Dic6
 Lower central C32 — C3×C6 — C12.30Dic6
 Upper central C1 — C2×C4 — C2×C8

Generators and relations for C12.30Dic6
G = < a,b,c | a12=1, b12=a6, c2=a9b6, ab=ba, cac-1=a5, cbc-1=a3b11 >

Subgroups: 308 in 114 conjugacy classes, 61 normal (23 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×3], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×12], C12 [×8], C2×C6 [×4], C42, C2×C8, C2×C8, C3×C6 [×3], C3⋊C8 [×4], C24 [×4], C2×Dic3 [×8], C2×C12 [×4], C4⋊C8, C3⋊Dic3 [×2], C3⋊Dic3, C3×C12 [×2], C62, C2×C3⋊C8 [×4], C4×Dic3 [×4], C2×C24 [×4], C324C8, C3×C24, C2×C3⋊Dic3 [×2], C6×C12, Dic3⋊C8 [×4], C2×C324C8, C4×C3⋊Dic3, C6×C24, C12.30Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C8 [×2], C2×C4, D4, Q8, D6 [×4], C4⋊C4, C2×C8, M4(2), C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C4⋊C8, C2×C3⋊S3, S3×C8 [×4], C8⋊S3 [×4], Dic3⋊C4 [×4], C324Q8, C4×C3⋊S3, C327D4, Dic3⋊C8 [×4], C8×C3⋊S3, C24⋊S3, C6.Dic6, C12.30Dic6

Smallest permutation representation of C12.30Dic6
Regular action on 288 points
Generators in S288
(1 77 276 118 222 253 13 89 288 106 234 241)(2 78 277 119 223 254 14 90 265 107 235 242)(3 79 278 120 224 255 15 91 266 108 236 243)(4 80 279 97 225 256 16 92 267 109 237 244)(5 81 280 98 226 257 17 93 268 110 238 245)(6 82 281 99 227 258 18 94 269 111 239 246)(7 83 282 100 228 259 19 95 270 112 240 247)(8 84 283 101 229 260 20 96 271 113 217 248)(9 85 284 102 230 261 21 73 272 114 218 249)(10 86 285 103 231 262 22 74 273 115 219 250)(11 87 286 104 232 263 23 75 274 116 220 251)(12 88 287 105 233 264 24 76 275 117 221 252)(25 186 134 69 156 194 37 174 122 57 168 206)(26 187 135 70 157 195 38 175 123 58 145 207)(27 188 136 71 158 196 39 176 124 59 146 208)(28 189 137 72 159 197 40 177 125 60 147 209)(29 190 138 49 160 198 41 178 126 61 148 210)(30 191 139 50 161 199 42 179 127 62 149 211)(31 192 140 51 162 200 43 180 128 63 150 212)(32 169 141 52 163 201 44 181 129 64 151 213)(33 170 142 53 164 202 45 182 130 65 152 214)(34 171 143 54 165 203 46 183 131 66 153 215)(35 172 144 55 166 204 47 184 132 67 154 216)(36 173 121 56 167 205 48 185 133 68 155 193)
(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)
(1 143 112 197)(2 214 113 136)(3 129 114 207)(4 200 115 122)(5 139 116 193)(6 210 117 132)(7 125 118 203)(8 196 119 142)(9 135 120 213)(10 206 97 128)(11 121 98 199)(12 216 99 138)(13 131 100 209)(14 202 101 124)(15 141 102 195)(16 212 103 134)(17 127 104 205)(18 198 105 144)(19 137 106 215)(20 208 107 130)(21 123 108 201)(22 194 109 140)(23 133 110 211)(24 204 111 126)(25 267 63 262)(26 255 64 284)(27 277 65 248)(28 241 66 270)(29 287 67 258)(30 251 68 280)(31 273 69 244)(32 261 70 266)(33 283 71 254)(34 247 72 276)(35 269 49 264)(36 257 50 286)(37 279 51 250)(38 243 52 272)(39 265 53 260)(40 253 54 282)(41 275 55 246)(42 263 56 268)(43 285 57 256)(44 249 58 278)(45 271 59 242)(46 259 60 288)(47 281 61 252)(48 245 62 274)(73 187 236 151)(74 168 237 180)(75 173 238 161)(76 154 239 190)(77 183 240 147)(78 164 217 176)(79 169 218 157)(80 150 219 186)(81 179 220 167)(82 160 221 172)(83 189 222 153)(84 146 223 182)(85 175 224 163)(86 156 225 192)(87 185 226 149)(88 166 227 178)(89 171 228 159)(90 152 229 188)(91 181 230 145)(92 162 231 174)(93 191 232 155)(94 148 233 184)(95 177 234 165)(96 158 235 170)

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6L 8A 8B 8C 8D 8E 8F 8G 8H 12A ··· 12P 24A ··· 24AF order 1 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 6 ··· 6 8 8 8 8 8 8 8 8 12 ··· 12 24 ··· 24 size 1 1 1 1 2 2 2 2 1 1 1 1 18 18 18 18 2 ··· 2 2 2 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + - + - image C1 C2 C2 C2 C4 C8 S3 D4 Q8 D6 M4(2) Dic6 C3⋊D4 C4×S3 S3×C8 C8⋊S3 kernel C12.30Dic6 C2×C32⋊4C8 C4×C3⋊Dic3 C6×C24 C2×C3⋊Dic3 C3⋊Dic3 C2×C24 C3×C12 C3×C12 C2×C12 C3×C6 C12 C12 C2×C6 C6 C6 # reps 1 1 1 1 4 8 4 1 1 4 2 8 8 8 16 16

Matrix representation of C12.30Dic6 in GL5(𝔽73)

 46 0 0 0 0 0 0 72 0 0 0 1 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 22 0 0 0 0 0 27 0 0 0 0 0 27 0 0 0 0 0 7 7 0 0 0 66 14
,
 1 0 0 0 0 0 2 62 0 0 0 60 71 0 0 0 0 0 61 22 0 0 0 10 12

G:=sub<GL(5,GF(73))| [46,0,0,0,0,0,0,1,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,1],[22,0,0,0,0,0,27,0,0,0,0,0,27,0,0,0,0,0,7,66,0,0,0,7,14],[1,0,0,0,0,0,2,60,0,0,0,62,71,0,0,0,0,0,61,10,0,0,0,22,12] >;

C12.30Dic6 in GAP, Magma, Sage, TeX

C_{12}._{30}{\rm Dic}_6
% in TeX

G:=Group("C12.30Dic6");
// GroupNames label

G:=SmallGroup(288,289);
// by ID

G=gap.SmallGroup(288,289);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,141,36,100,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=1,b^12=a^6,c^2=a^9*b^6,a*b=b*a,c*a*c^-1=a^5,c*b*c^-1=a^3*b^11>;
// generators/relations

׿
×
𝔽