Copied to
clipboard

G = C62.160D6order 432 = 24·33

44th non-split extension by C62 of D6 acting via D6/C6=C2

metabelian, supersoluble, monomial

Aliases: C62.160D6, (C6×C12)⋊13S3, (C3×C12).201D6, C3329(C4○D4), C3315D47C2, C338Q811C2, C3312D411C2, C35(C12.59D6), C3227(C4○D12), (C3×C62).69C22, (C32×C6).98C23, C335C4.20C22, (C32×C12).101C22, (C3×C6×C12)⋊6C2, (C2×C12)⋊4(C3⋊S3), C12.78(C2×C3⋊S3), (C4×C33⋊C2)⋊9C2, C6.42(C22×C3⋊S3), (C2×C4)⋊3(C33⋊C2), C4.16(C2×C33⋊C2), (C3×C6).187(C22×S3), C22.2(C2×C33⋊C2), C2.5(C22×C33⋊C2), (C2×C33⋊C2).18C22, (C2×C6).49(C2×C3⋊S3), SmallGroup(432,723)

Series: Derived Chief Lower central Upper central

C1C32×C6 — C62.160D6
C1C3C32C33C32×C6C2×C33⋊C2C4×C33⋊C2 — C62.160D6
C33C32×C6 — C62.160D6
C1C4C2×C4

Generators and relations for C62.160D6
 G = < a,b,c,d | a6=b6=1, c6=d2=b3, ab=ba, ac=ca, dad-1=a-1b3, bc=cb, dbd-1=b-1, dcd-1=c5 >

Subgroups: 2984 in 560 conjugacy classes, 179 normal (15 characteristic)
C1, C2, C2 [×3], C3 [×13], C4 [×2], C4 [×2], C22, C22 [×2], S3 [×26], C6 [×13], C6 [×13], C2×C4, C2×C4 [×2], D4 [×3], Q8, C32 [×13], Dic3 [×26], C12 [×26], D6 [×26], C2×C6 [×13], C4○D4, C3⋊S3 [×26], C3×C6 [×13], C3×C6 [×13], Dic6 [×13], C4×S3 [×26], D12 [×13], C3⋊D4 [×26], C2×C12 [×13], C33, C3⋊Dic3 [×26], C3×C12 [×26], C2×C3⋊S3 [×26], C62 [×13], C4○D12 [×13], C33⋊C2 [×2], C32×C6, C32×C6, C324Q8 [×13], C4×C3⋊S3 [×26], C12⋊S3 [×13], C327D4 [×26], C6×C12 [×13], C335C4 [×2], C32×C12 [×2], C2×C33⋊C2 [×2], C3×C62, C12.59D6 [×13], C338Q8, C4×C33⋊C2 [×2], C3312D4, C3315D4 [×2], C3×C6×C12, C62.160D6
Quotients: C1, C2 [×7], C22 [×7], S3 [×13], C23, D6 [×39], C4○D4, C3⋊S3 [×13], C22×S3 [×13], C2×C3⋊S3 [×39], C4○D12 [×13], C33⋊C2, C22×C3⋊S3 [×13], C2×C33⋊C2 [×3], C12.59D6 [×13], C22×C33⋊C2, C62.160D6

Smallest permutation representation of C62.160D6
On 216 points
Generators in S216
(1 107 114)(2 108 115)(3 97 116)(4 98 117)(5 99 118)(6 100 119)(7 101 120)(8 102 109)(9 103 110)(10 104 111)(11 105 112)(12 106 113)(13 81 25)(14 82 26)(15 83 27)(16 84 28)(17 73 29)(18 74 30)(19 75 31)(20 76 32)(21 77 33)(22 78 34)(23 79 35)(24 80 36)(37 125 90 43 131 96)(38 126 91 44 132 85)(39 127 92 45 121 86)(40 128 93 46 122 87)(41 129 94 47 123 88)(42 130 95 48 124 89)(49 149 179 55 155 173)(50 150 180 56 156 174)(51 151 169 57 145 175)(52 152 170 58 146 176)(53 153 171 59 147 177)(54 154 172 60 148 178)(61 195 144)(62 196 133)(63 197 134)(64 198 135)(65 199 136)(66 200 137)(67 201 138)(68 202 139)(69 203 140)(70 204 141)(71 193 142)(72 194 143)(157 188 216 163 182 210)(158 189 205 164 183 211)(159 190 206 165 184 212)(160 191 207 166 185 213)(161 192 208 167 186 214)(162 181 209 168 187 215)
(1 62 75 7 68 81)(2 63 76 8 69 82)(3 64 77 9 70 83)(4 65 78 10 71 84)(5 66 79 11 72 73)(6 67 80 12 61 74)(13 114 133 19 120 139)(14 115 134 20 109 140)(15 116 135 21 110 141)(16 117 136 22 111 142)(17 118 137 23 112 143)(18 119 138 24 113 144)(25 107 196 31 101 202)(26 108 197 32 102 203)(27 97 198 33 103 204)(28 98 199 34 104 193)(29 99 200 35 105 194)(30 100 201 36 106 195)(37 213 170 43 207 176)(38 214 171 44 208 177)(39 215 172 45 209 178)(40 216 173 46 210 179)(41 205 174 47 211 180)(42 206 175 48 212 169)(49 122 157 55 128 163)(50 123 158 56 129 164)(51 124 159 57 130 165)(52 125 160 58 131 166)(53 126 161 59 132 167)(54 127 162 60 121 168)(85 186 153 91 192 147)(86 187 154 92 181 148)(87 188 155 93 182 149)(88 189 156 94 183 150)(89 190 145 95 184 151)(90 191 146 96 185 152)
(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)
(1 124 7 130)(2 129 8 123)(3 122 9 128)(4 127 10 121)(5 132 11 126)(6 125 12 131)(13 184 19 190)(14 189 20 183)(15 182 21 188)(16 187 22 181)(17 192 23 186)(18 185 24 191)(25 206 31 212)(26 211 32 205)(27 216 33 210)(28 209 34 215)(29 214 35 208)(30 207 36 213)(37 100 43 106)(38 105 44 99)(39 98 45 104)(40 103 46 97)(41 108 47 102)(42 101 48 107)(49 70 55 64)(50 63 56 69)(51 68 57 62)(52 61 58 67)(53 66 59 72)(54 71 60 65)(73 167 79 161)(74 160 80 166)(75 165 81 159)(76 158 82 164)(77 163 83 157)(78 168 84 162)(85 118 91 112)(86 111 92 117)(87 116 93 110)(88 109 94 115)(89 114 95 120)(90 119 96 113)(133 145 139 151)(134 150 140 156)(135 155 141 149)(136 148 142 154)(137 153 143 147)(138 146 144 152)(169 202 175 196)(170 195 176 201)(171 200 177 194)(172 193 178 199)(173 198 179 204)(174 203 180 197)

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

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

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

114 conjugacy classes

class 1 2A2B2C2D3A···3M4A4B4C4D4E6A···6AM12A···12AZ
order122223···3444446···612···12
size11254542···211254542···22···2

114 irreducible representations

dim11111122222
type+++++++++
imageC1C2C2C2C2C2S3D6D6C4○D4C4○D12
kernelC62.160D6C338Q8C4×C33⋊C2C3312D4C3315D4C3×C6×C12C6×C12C3×C12C62C33C32
# reps112121132613252

Matrix representation of C62.160D6 in GL6(𝔽13)

300000
040000
001000
0001200
000010
0000012
,
1000000
040000
0010000
000400
0000100
000004
,
600000
020000
006000
000200
000050
000005
,
020000
600000
000200
006000
000005
000050

G:=sub<GL(6,GF(13))| [3,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,12],[10,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,4,0,0,0,0,0,0,10,0,0,0,0,0,0,4],[6,0,0,0,0,0,0,2,0,0,0,0,0,0,6,0,0,0,0,0,0,2,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,6,0,0,0,0,2,0,0,0,0,0,0,0,0,5,0,0,0,0,5,0] >;

C62.160D6 in GAP, Magma, Sage, TeX

C_6^2._{160}D_6
% in TeX

G:=Group("C6^2.160D6");
// GroupNames label

G:=SmallGroup(432,723);
// by ID

G=gap.SmallGroup(432,723);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,64,254,1124,4037,14118]);
// Polycyclic

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

׿
×
𝔽