metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.17SD16, C42.222D6, C3⋊C8⋊8Q8, C4⋊Q8.6S3, C4⋊C4.80D6, C3⋊3(C8⋊3Q8), C4.35(S3×Q8), C6.31(C4⋊Q8), C12.36(C2×Q8), (C2×C12).153D4, C4.4(D4.S3), C6.60(C2×SD16), C12⋊2Q8.20C2, C4.4(Q8⋊2S3), (C4×C12).127C22, (C2×C12).398C23, C12.Q8.16C2, C4⋊Dic3.158C22, C2.11(Dic3⋊Q8), (C4×C3⋊C8).13C2, (C3×C4⋊Q8).6C2, (C2×C6).529(C2×D4), C2.14(C2×D4.S3), (C2×C3⋊C8).261C22, C2.12(C2×Q8⋊2S3), (C2×C4).135(C3⋊D4), (C3×C4⋊C4).127C22, (C2×C4).495(C22×S3), C22.201(C2×C3⋊D4), SmallGroup(192,639)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.SD16
G = < a,b,c | a12=b8=1, c2=a6, bab-1=a5, cac-1=a7, cbc-1=b3 >
Subgroups: 240 in 98 conjugacy classes, 51 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, C4.Q8, C4⋊Q8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C4⋊C4, C3×C4⋊C4, C2×Dic6, C6×Q8, C8⋊3Q8, C4×C3⋊C8, C12.Q8, C12⋊2Q8, C3×C4⋊Q8, C12.SD16
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, SD16, C2×D4, C2×Q8, C3⋊D4, C22×S3, C4⋊Q8, C2×SD16, D4.S3, Q8⋊2S3, S3×Q8, C2×C3⋊D4, C8⋊3Q8, C2×D4.S3, C2×Q8⋊2S3, Dic3⋊Q8, C12.SD16
►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)
(1 147 127 74 163 49 173 120)(2 152 128 79 164 54 174 113)(3 145 129 84 165 59 175 118)(4 150 130 77 166 52 176 111)(5 155 131 82 167 57 177 116)(6 148 132 75 168 50 178 109)(7 153 121 80 157 55 179 114)(8 146 122 73 158 60 180 119)(9 151 123 78 159 53 169 112)(10 156 124 83 160 58 170 117)(11 149 125 76 161 51 171 110)(12 154 126 81 162 56 172 115)(13 138 108 33 45 87 191 71)(14 143 97 26 46 92 192 64)(15 136 98 31 47 85 181 69)(16 141 99 36 48 90 182 62)(17 134 100 29 37 95 183 67)(18 139 101 34 38 88 184 72)(19 144 102 27 39 93 185 65)(20 137 103 32 40 86 186 70)(21 142 104 25 41 91 187 63)(22 135 105 30 42 96 188 68)(23 140 106 35 43 89 189 61)(24 133 107 28 44 94 190 66)
(1 48 7 42)(2 43 8 37)(3 38 9 44)(4 45 10 39)(5 40 11 46)(6 47 12 41)(13 160 19 166)(14 167 20 161)(15 162 21 168)(16 157 22 163)(17 164 23 158)(18 159 24 165)(25 50 31 56)(26 57 32 51)(27 52 33 58)(28 59 34 53)(29 54 35 60)(30 49 36 55)(61 146 67 152)(62 153 68 147)(63 148 69 154)(64 155 70 149)(65 150 71 156)(66 145 72 151)(73 95 79 89)(74 90 80 96)(75 85 81 91)(76 92 82 86)(77 87 83 93)(78 94 84 88)(97 131 103 125)(98 126 104 132)(99 121 105 127)(100 128 106 122)(101 123 107 129)(102 130 108 124)(109 136 115 142)(110 143 116 137)(111 138 117 144)(112 133 118 139)(113 140 119 134)(114 135 120 141)(169 190 175 184)(170 185 176 191)(171 192 177 186)(172 187 178 181)(173 182 179 188)(174 189 180 183)
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), (1,147,127,74,163,49,173,120)(2,152,128,79,164,54,174,113)(3,145,129,84,165,59,175,118)(4,150,130,77,166,52,176,111)(5,155,131,82,167,57,177,116)(6,148,132,75,168,50,178,109)(7,153,121,80,157,55,179,114)(8,146,122,73,158,60,180,119)(9,151,123,78,159,53,169,112)(10,156,124,83,160,58,170,117)(11,149,125,76,161,51,171,110)(12,154,126,81,162,56,172,115)(13,138,108,33,45,87,191,71)(14,143,97,26,46,92,192,64)(15,136,98,31,47,85,181,69)(16,141,99,36,48,90,182,62)(17,134,100,29,37,95,183,67)(18,139,101,34,38,88,184,72)(19,144,102,27,39,93,185,65)(20,137,103,32,40,86,186,70)(21,142,104,25,41,91,187,63)(22,135,105,30,42,96,188,68)(23,140,106,35,43,89,189,61)(24,133,107,28,44,94,190,66), (1,48,7,42)(2,43,8,37)(3,38,9,44)(4,45,10,39)(5,40,11,46)(6,47,12,41)(13,160,19,166)(14,167,20,161)(15,162,21,168)(16,157,22,163)(17,164,23,158)(18,159,24,165)(25,50,31,56)(26,57,32,51)(27,52,33,58)(28,59,34,53)(29,54,35,60)(30,49,36,55)(61,146,67,152)(62,153,68,147)(63,148,69,154)(64,155,70,149)(65,150,71,156)(66,145,72,151)(73,95,79,89)(74,90,80,96)(75,85,81,91)(76,92,82,86)(77,87,83,93)(78,94,84,88)(97,131,103,125)(98,126,104,132)(99,121,105,127)(100,128,106,122)(101,123,107,129)(102,130,108,124)(109,136,115,142)(110,143,116,137)(111,138,117,144)(112,133,118,139)(113,140,119,134)(114,135,120,141)(169,190,175,184)(170,185,176,191)(171,192,177,186)(172,187,178,181)(173,182,179,188)(174,189,180,183)>;
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), (1,147,127,74,163,49,173,120)(2,152,128,79,164,54,174,113)(3,145,129,84,165,59,175,118)(4,150,130,77,166,52,176,111)(5,155,131,82,167,57,177,116)(6,148,132,75,168,50,178,109)(7,153,121,80,157,55,179,114)(8,146,122,73,158,60,180,119)(9,151,123,78,159,53,169,112)(10,156,124,83,160,58,170,117)(11,149,125,76,161,51,171,110)(12,154,126,81,162,56,172,115)(13,138,108,33,45,87,191,71)(14,143,97,26,46,92,192,64)(15,136,98,31,47,85,181,69)(16,141,99,36,48,90,182,62)(17,134,100,29,37,95,183,67)(18,139,101,34,38,88,184,72)(19,144,102,27,39,93,185,65)(20,137,103,32,40,86,186,70)(21,142,104,25,41,91,187,63)(22,135,105,30,42,96,188,68)(23,140,106,35,43,89,189,61)(24,133,107,28,44,94,190,66), (1,48,7,42)(2,43,8,37)(3,38,9,44)(4,45,10,39)(5,40,11,46)(6,47,12,41)(13,160,19,166)(14,167,20,161)(15,162,21,168)(16,157,22,163)(17,164,23,158)(18,159,24,165)(25,50,31,56)(26,57,32,51)(27,52,33,58)(28,59,34,53)(29,54,35,60)(30,49,36,55)(61,146,67,152)(62,153,68,147)(63,148,69,154)(64,155,70,149)(65,150,71,156)(66,145,72,151)(73,95,79,89)(74,90,80,96)(75,85,81,91)(76,92,82,86)(77,87,83,93)(78,94,84,88)(97,131,103,125)(98,126,104,132)(99,121,105,127)(100,128,106,122)(101,123,107,129)(102,130,108,124)(109,136,115,142)(110,143,116,137)(111,138,117,144)(112,133,118,139)(113,140,119,134)(114,135,120,141)(169,190,175,184)(170,185,176,191)(171,192,177,186)(172,187,178,181)(173,182,179,188)(174,189,180,183) );
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)], [(1,147,127,74,163,49,173,120),(2,152,128,79,164,54,174,113),(3,145,129,84,165,59,175,118),(4,150,130,77,166,52,176,111),(5,155,131,82,167,57,177,116),(6,148,132,75,168,50,178,109),(7,153,121,80,157,55,179,114),(8,146,122,73,158,60,180,119),(9,151,123,78,159,53,169,112),(10,156,124,83,160,58,170,117),(11,149,125,76,161,51,171,110),(12,154,126,81,162,56,172,115),(13,138,108,33,45,87,191,71),(14,143,97,26,46,92,192,64),(15,136,98,31,47,85,181,69),(16,141,99,36,48,90,182,62),(17,134,100,29,37,95,183,67),(18,139,101,34,38,88,184,72),(19,144,102,27,39,93,185,65),(20,137,103,32,40,86,186,70),(21,142,104,25,41,91,187,63),(22,135,105,30,42,96,188,68),(23,140,106,35,43,89,189,61),(24,133,107,28,44,94,190,66)], [(1,48,7,42),(2,43,8,37),(3,38,9,44),(4,45,10,39),(5,40,11,46),(6,47,12,41),(13,160,19,166),(14,167,20,161),(15,162,21,168),(16,157,22,163),(17,164,23,158),(18,159,24,165),(25,50,31,56),(26,57,32,51),(27,52,33,58),(28,59,34,53),(29,54,35,60),(30,49,36,55),(61,146,67,152),(62,153,68,147),(63,148,69,154),(64,155,70,149),(65,150,71,156),(66,145,72,151),(73,95,79,89),(74,90,80,96),(75,85,81,91),(76,92,82,86),(77,87,83,93),(78,94,84,88),(97,131,103,125),(98,126,104,132),(99,121,105,127),(100,128,106,122),(101,123,107,129),(102,130,108,124),(109,136,115,142),(110,143,116,137),(111,138,117,144),(112,133,118,139),(113,140,119,134),(114,135,120,141),(169,190,175,184),(170,185,176,191),(171,192,177,186),(172,187,178,181),(173,182,179,188),(174,189,180,183)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12F | 12G | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 6 | ··· | 6 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | Q8 | D4 | D6 | D6 | SD16 | C3⋊D4 | D4.S3 | Q8⋊2S3 | S3×Q8 |
| kernel | C12.SD16 | C4×C3⋊C8 | C12.Q8 | C12⋊2Q8 | C3×C4⋊Q8 | C4⋊Q8 | C3⋊C8 | C2×C12 | C42 | C4⋊C4 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 1 | 4 | 1 | 1 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 2 | 2 |
Matrix representation of C12.SD16 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 55 | 50 | 0 | 0 | 0 | 0 |
| 68 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 69 | 61 | 0 | 0 |
| 0 | 0 | 0 | 0 | 67 | 6 |
| 0 | 0 | 0 | 0 | 67 | 67 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 21 | 64 | 0 | 0 |
| 0 | 0 | 57 | 52 | 0 | 0 |
| 0 | 0 | 0 | 0 | 43 | 62 |
| 0 | 0 | 0 | 0 | 62 | 30 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0],[55,68,0,0,0,0,50,18,0,0,0,0,0,0,0,69,0,0,0,0,18,61,0,0,0,0,0,0,67,67,0,0,0,0,6,67],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,21,57,0,0,0,0,64,52,0,0,0,0,0,0,43,62,0,0,0,0,62,30] >;
C12.SD16 in GAP, Magma, Sage, TeX
C_{12}.{\rm SD}_{16} % in TeX
G:=Group("C12.SD16"); // GroupNames label
G:=SmallGroup(192,639);
// by ID
G=gap.SmallGroup(192,639);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,64,1094,135,58,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^3>;
// generators/relations