Copied to
clipboard

G = C12.10Dic6order 288 = 25·32

10th non-split extension by C12 of Dic6 acting via Dic6/C6=C22

metabelian, supersoluble, monomial

Aliases: C12.10Dic6, C62.112D4, C12.25(C4×S3), C324C84C4, (C2×C12).86D6, (C3×C12).12Q8, C328(C4.Q8), (C3×C6).28SD16, C6.11(D4.S3), (C6×C12).53C22, C33(C12.Q8), C4.2(C324Q8), C6.17(Dic3⋊C4), C6.11(Q82S3), C12⋊Dic3.13C2, C2.1(C329SD16), C2.1(C3211SD16), C2.4(C6.Dic6), C22.13(C327D4), C4.12(C4×C3⋊S3), C4⋊C4.2(C3⋊S3), (C3×C4⋊C4).18S3, (C3×C6).38(C4⋊C4), (C3×C12).47(C2×C4), (C32×C4⋊C4).5C2, (C2×C6).88(C3⋊D4), (C2×C324C8).6C2, (C2×C4).35(C2×C3⋊S3), SmallGroup(288,283)

Series: Derived Chief Lower central Upper central

C1C3×C12 — C12.10Dic6
C1C3C32C3×C6C62C6×C12C2×C324C8 — C12.10Dic6
C32C3×C6C3×C12 — C12.10Dic6
C1C22C2×C4C4⋊C4

Generators and relations for C12.10Dic6
 G = < a,b,c | a12=b12=1, c2=a9b6, bab-1=a7, cac-1=a5, cbc-1=a3b-1 >

Subgroups: 332 in 108 conjugacy classes, 59 normal (21 characteristic)
C1, C2 [×3], C3 [×4], C4 [×2], C4 [×2], C22, C6 [×12], C8 [×2], C2×C4, C2×C4 [×2], C32, Dic3 [×4], C12 [×8], C12 [×4], C2×C6 [×4], C4⋊C4, C4⋊C4, C2×C8, C3×C6 [×3], C3⋊C8 [×8], C2×Dic3 [×4], C2×C12 [×4], C2×C12 [×4], C4.Q8, C3⋊Dic3, C3×C12 [×2], C3×C12, C62, C2×C3⋊C8 [×4], C4⋊Dic3 [×4], C3×C4⋊C4 [×4], C324C8 [×2], C2×C3⋊Dic3, C6×C12, C6×C12, C12.Q8 [×4], C2×C324C8, C12⋊Dic3, C32×C4⋊C4, C12.10Dic6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×4], C2×C4, D4, Q8, D6 [×4], C4⋊C4, SD16 [×2], C3⋊S3, Dic6 [×4], C4×S3 [×4], C3⋊D4 [×4], C4.Q8, C2×C3⋊S3, Dic3⋊C4 [×4], D4.S3 [×4], Q82S3 [×4], C324Q8, C4×C3⋊S3, C327D4, C12.Q8 [×4], C6.Dic6, C329SD16, C3211SD16, C12.10Dic6

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

dim111112222222244
type+++++-++--+
imageC1C2C2C2C4S3Q8D4D6SD16Dic6C4×S3C3⋊D4D4.S3Q82S3
kernelC12.10Dic6C2×C324C8C12⋊Dic3C32×C4⋊C4C324C8C3×C4⋊C4C3×C12C62C2×C12C3×C6C12C12C2×C6C6C6
# reps111144114488844

Matrix representation of C12.10Dic6 in GL6(𝔽73)

100000
010000
000100
00727200
0000722
0000721
,
1720000
100000
0014700
0066700
00004151
00003032
,
44540000
25290000
00191400
00685400
0000012
00006712

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,2,1],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,14,66,0,0,0,0,7,7,0,0,0,0,0,0,41,30,0,0,0,0,51,32],[44,25,0,0,0,0,54,29,0,0,0,0,0,0,19,68,0,0,0,0,14,54,0,0,0,0,0,0,0,67,0,0,0,0,12,12] >;

C12.10Dic6 in GAP, Magma, Sage, TeX

C_{12}._{10}{\rm Dic}_6
% in TeX

G:=Group("C12.10Dic6");
// GroupNames label

G:=SmallGroup(288,283);
// by ID

G=gap.SmallGroup(288,283);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,365,36,346,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c|a^12=b^12=1,c^2=a^9*b^6,b*a*b^-1=a^7,c*a*c^-1=a^5,c*b*c^-1=a^3*b^-1>;
// generators/relations

׿
×
𝔽