metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊7Q16, Q8.7D12, C42.60D6, C4⋊C4.257D6, C4⋊3(C3⋊Q16), C12.22(C2×D4), (C2×C12).68D4, C4.18(C2×D12), C3⋊3(C4⋊2Q16), (Q8×C12).9C2, (C3×Q8).19D4, (C4×Q8).15S3, C6.36(C2×Q16), (C2×Q8).188D6, C12⋊C8.19C2, C12.64(C4○D4), C4.14(C4○D12), C6.69(C4⋊D4), C12⋊2Q8.15C2, C2.17(C12⋊7D4), (C2×C12).351C23, (C4×C12).102C22, C6.SD16.11C2, (C6×Q8).199C22, C2.10(Q8.14D6), C6.112(C8.C22), (C2×Dic6).102C22, C2.7(C2×C3⋊Q16), (C2×C6).482(C2×D4), (C2×C3⋊Q16).5C2, (C2×C3⋊C8).104C22, (C2×C4).250(C3⋊D4), (C3×C4⋊C4).288C22, (C2×C4).451(C22×S3), C22.157(C2×C3⋊D4), SmallGroup(192,590)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊7Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a-1, ac=ca, cbc-1=b-1 >
Subgroups: 264 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C3, C4, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Q8, Dic3, C12, C12, C12, C2×C6, C42, C42, C4⋊C4, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, Q8⋊C4, C4⋊C8, C4×Q8, C4⋊Q8, C2×Q16, C2×C3⋊C8, C4⋊Dic3, C3⋊Q16, C4×C12, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C4⋊2Q16, C12⋊C8, C6.SD16, C12⋊2Q8, C2×C3⋊Q16, Q8×C12, C12⋊7Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×Q16, C8.C22, C3⋊Q16, C2×D12, C4○D12, C2×C3⋊D4, C4⋊2Q16, C12⋊7D4, C2×C3⋊Q16, Q8.14D6, C12⋊7Q16
►Regular action on 192 points
Generators in S192
(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)
(1 144 93 33 159 64 111 126)(2 143 94 32 160 63 112 125)(3 142 95 31 161 62 113 124)(4 141 96 30 162 61 114 123)(5 140 85 29 163 72 115 122)(6 139 86 28 164 71 116 121)(7 138 87 27 165 70 117 132)(8 137 88 26 166 69 118 131)(9 136 89 25 167 68 119 130)(10 135 90 36 168 67 120 129)(11 134 91 35 157 66 109 128)(12 133 92 34 158 65 110 127)(13 155 46 105 80 174 188 53)(14 154 47 104 81 173 189 52)(15 153 48 103 82 172 190 51)(16 152 37 102 83 171 191 50)(17 151 38 101 84 170 192 49)(18 150 39 100 73 169 181 60)(19 149 40 99 74 180 182 59)(20 148 41 98 75 179 183 58)(21 147 42 97 76 178 184 57)(22 146 43 108 77 177 185 56)(23 145 44 107 78 176 186 55)(24 156 45 106 79 175 187 54)
(1 59 159 99)(2 60 160 100)(3 49 161 101)(4 50 162 102)(5 51 163 103)(6 52 164 104)(7 53 165 105)(8 54 166 106)(9 55 167 107)(10 56 168 108)(11 57 157 97)(12 58 158 98)(13 27 80 132)(14 28 81 121)(15 29 82 122)(16 30 83 123)(17 31 84 124)(18 32 73 125)(19 33 74 126)(20 34 75 127)(21 35 76 128)(22 36 77 129)(23 25 78 130)(24 26 79 131)(37 141 191 61)(38 142 192 62)(39 143 181 63)(40 144 182 64)(41 133 183 65)(42 134 184 66)(43 135 185 67)(44 136 186 68)(45 137 187 69)(46 138 188 70)(47 139 189 71)(48 140 190 72)(85 172 115 153)(86 173 116 154)(87 174 117 155)(88 175 118 156)(89 176 119 145)(90 177 120 146)(91 178 109 147)(92 179 110 148)(93 180 111 149)(94 169 112 150)(95 170 113 151)(96 171 114 152)
G:=sub<Sym(192)| (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), (1,144,93,33,159,64,111,126)(2,143,94,32,160,63,112,125)(3,142,95,31,161,62,113,124)(4,141,96,30,162,61,114,123)(5,140,85,29,163,72,115,122)(6,139,86,28,164,71,116,121)(7,138,87,27,165,70,117,132)(8,137,88,26,166,69,118,131)(9,136,89,25,167,68,119,130)(10,135,90,36,168,67,120,129)(11,134,91,35,157,66,109,128)(12,133,92,34,158,65,110,127)(13,155,46,105,80,174,188,53)(14,154,47,104,81,173,189,52)(15,153,48,103,82,172,190,51)(16,152,37,102,83,171,191,50)(17,151,38,101,84,170,192,49)(18,150,39,100,73,169,181,60)(19,149,40,99,74,180,182,59)(20,148,41,98,75,179,183,58)(21,147,42,97,76,178,184,57)(22,146,43,108,77,177,185,56)(23,145,44,107,78,176,186,55)(24,156,45,106,79,175,187,54), (1,59,159,99)(2,60,160,100)(3,49,161,101)(4,50,162,102)(5,51,163,103)(6,52,164,104)(7,53,165,105)(8,54,166,106)(9,55,167,107)(10,56,168,108)(11,57,157,97)(12,58,158,98)(13,27,80,132)(14,28,81,121)(15,29,82,122)(16,30,83,123)(17,31,84,124)(18,32,73,125)(19,33,74,126)(20,34,75,127)(21,35,76,128)(22,36,77,129)(23,25,78,130)(24,26,79,131)(37,141,191,61)(38,142,192,62)(39,143,181,63)(40,144,182,64)(41,133,183,65)(42,134,184,66)(43,135,185,67)(44,136,186,68)(45,137,187,69)(46,138,188,70)(47,139,189,71)(48,140,190,72)(85,172,115,153)(86,173,116,154)(87,174,117,155)(88,175,118,156)(89,176,119,145)(90,177,120,146)(91,178,109,147)(92,179,110,148)(93,180,111,149)(94,169,112,150)(95,170,113,151)(96,171,114,152)>;
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), (1,144,93,33,159,64,111,126)(2,143,94,32,160,63,112,125)(3,142,95,31,161,62,113,124)(4,141,96,30,162,61,114,123)(5,140,85,29,163,72,115,122)(6,139,86,28,164,71,116,121)(7,138,87,27,165,70,117,132)(8,137,88,26,166,69,118,131)(9,136,89,25,167,68,119,130)(10,135,90,36,168,67,120,129)(11,134,91,35,157,66,109,128)(12,133,92,34,158,65,110,127)(13,155,46,105,80,174,188,53)(14,154,47,104,81,173,189,52)(15,153,48,103,82,172,190,51)(16,152,37,102,83,171,191,50)(17,151,38,101,84,170,192,49)(18,150,39,100,73,169,181,60)(19,149,40,99,74,180,182,59)(20,148,41,98,75,179,183,58)(21,147,42,97,76,178,184,57)(22,146,43,108,77,177,185,56)(23,145,44,107,78,176,186,55)(24,156,45,106,79,175,187,54), (1,59,159,99)(2,60,160,100)(3,49,161,101)(4,50,162,102)(5,51,163,103)(6,52,164,104)(7,53,165,105)(8,54,166,106)(9,55,167,107)(10,56,168,108)(11,57,157,97)(12,58,158,98)(13,27,80,132)(14,28,81,121)(15,29,82,122)(16,30,83,123)(17,31,84,124)(18,32,73,125)(19,33,74,126)(20,34,75,127)(21,35,76,128)(22,36,77,129)(23,25,78,130)(24,26,79,131)(37,141,191,61)(38,142,192,62)(39,143,181,63)(40,144,182,64)(41,133,183,65)(42,134,184,66)(43,135,185,67)(44,136,186,68)(45,137,187,69)(46,138,188,70)(47,139,189,71)(48,140,190,72)(85,172,115,153)(86,173,116,154)(87,174,117,155)(88,175,118,156)(89,176,119,145)(90,177,120,146)(91,178,109,147)(92,179,110,148)(93,180,111,149)(94,169,112,150)(95,170,113,151)(96,171,114,152) );
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)], [(1,144,93,33,159,64,111,126),(2,143,94,32,160,63,112,125),(3,142,95,31,161,62,113,124),(4,141,96,30,162,61,114,123),(5,140,85,29,163,72,115,122),(6,139,86,28,164,71,116,121),(7,138,87,27,165,70,117,132),(8,137,88,26,166,69,118,131),(9,136,89,25,167,68,119,130),(10,135,90,36,168,67,120,129),(11,134,91,35,157,66,109,128),(12,133,92,34,158,65,110,127),(13,155,46,105,80,174,188,53),(14,154,47,104,81,173,189,52),(15,153,48,103,82,172,190,51),(16,152,37,102,83,171,191,50),(17,151,38,101,84,170,192,49),(18,150,39,100,73,169,181,60),(19,149,40,99,74,180,182,59),(20,148,41,98,75,179,183,58),(21,147,42,97,76,178,184,57),(22,146,43,108,77,177,185,56),(23,145,44,107,78,176,186,55),(24,156,45,106,79,175,187,54)], [(1,59,159,99),(2,60,160,100),(3,49,161,101),(4,50,162,102),(5,51,163,103),(6,52,164,104),(7,53,165,105),(8,54,166,106),(9,55,167,107),(10,56,168,108),(11,57,157,97),(12,58,158,98),(13,27,80,132),(14,28,81,121),(15,29,82,122),(16,30,83,123),(17,31,84,124),(18,32,73,125),(19,33,74,126),(20,34,75,127),(21,35,76,128),(22,36,77,129),(23,25,78,130),(24,26,79,131),(37,141,191,61),(38,142,192,62),(39,143,181,63),(40,144,182,64),(41,133,183,65),(42,134,184,66),(43,135,185,67),(44,136,186,68),(45,137,187,69),(46,138,188,70),(47,139,189,71),(48,140,190,72),(85,172,115,153),(86,173,116,154),(87,174,117,155),(88,175,118,156),(89,176,119,145),(90,177,120,146),(91,178,109,147),(92,179,110,148),(93,180,111,149),(94,169,112,150),(95,170,113,151),(96,171,114,152)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | ··· | 4I | 4J | 4K | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 24 | 24 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | - | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | Q16 | C4○D4 | C3⋊D4 | D12 | C4○D12 | C8.C22 | C3⋊Q16 | Q8.14D6 |
| kernel | C12⋊7Q16 | C12⋊C8 | C6.SD16 | C12⋊2Q8 | C2×C3⋊Q16 | Q8×C12 | C4×Q8 | C2×C12 | C3×Q8 | C42 | C4⋊C4 | C2×Q8 | C12 | C12 | C2×C4 | Q8 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of C12⋊7Q16 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 47 | 39 | 0 | 0 |
| 65 | 26 | 0 | 0 |
| 0 | 0 | 0 | 41 |
| 0 | 0 | 16 | 41 |
| 43 | 13 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 0 | 0 | 1 | 16 |
| 0 | 0 | 9 | 72 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,72,0,0,0,0,72],[47,65,0,0,39,26,0,0,0,0,0,16,0,0,41,41],[43,60,0,0,13,30,0,0,0,0,1,9,0,0,16,72] >;
C12⋊7Q16 in GAP, Magma, Sage, TeX
C_{12}\rtimes_7Q_{16} % in TeX
G:=Group("C12:7Q16"); // GroupNames label
G:=SmallGroup(192,590);
// by ID
G=gap.SmallGroup(192,590);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations