metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12⋊3Q16, C42.225D6, C3⋊C8.18D4, C4.19(S3×D4), C4⋊Q8.11S3, C12.40(C2×D4), C4⋊1(C3⋊Q16), C3⋊2(C4⋊Q16), (C2×Q8).72D6, C6.44(C2×Q16), (C2×C12).161D4, C6.25(C4⋊1D4), C12⋊2Q8.21C2, (C6×Q8).66C22, C2.16(C12⋊3D4), (C2×C12).410C23, (C4×C12).139C22, (C2×Dic6).115C22, (C4×C3⋊C8).14C2, (C3×C4⋊Q8).11C2, (C2×C6).541(C2×D4), (C2×C3⋊Q16).7C2, C2.15(C2×C3⋊Q16), (C2×C3⋊C8).264C22, (C2×C4).138(C3⋊D4), (C2×C4).507(C22×S3), C22.213(C2×C3⋊D4), SmallGroup(192,651)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12⋊3Q16
G = < a,b,c | a12=b8=1, c2=b4, bab-1=a5, cac-1=a7, cbc-1=b-1 >
Subgroups: 304 in 122 conjugacy classes, 51 normal (15 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C4×C8, C4⋊Q8, C4⋊Q8, C2×Q16, C2×C3⋊C8, C4⋊Dic3, C3⋊Q16, C4×C12, C3×C4⋊C4, C2×Dic6, C6×Q8, C4⋊Q16, C4×C3⋊C8, C12⋊2Q8, C2×C3⋊Q16, C3×C4⋊Q8, C12⋊3Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×Q16, C3⋊Q16, S3×D4, C2×C3⋊D4, C4⋊Q16, C12⋊3D4, C2×C3⋊Q16, C12⋊3Q16
►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 98 115 16 170 126 61 191)(2 103 116 21 171 131 62 184)(3 108 117 14 172 124 63 189)(4 101 118 19 173 129 64 182)(5 106 119 24 174 122 65 187)(6 99 120 17 175 127 66 192)(7 104 109 22 176 132 67 185)(8 97 110 15 177 125 68 190)(9 102 111 20 178 130 69 183)(10 107 112 13 179 123 70 188)(11 100 113 18 180 128 71 181)(12 105 114 23 169 121 72 186)(25 54 77 164 140 96 38 150)(26 59 78 157 141 89 39 155)(27 52 79 162 142 94 40 148)(28 57 80 167 143 87 41 153)(29 50 81 160 144 92 42 146)(30 55 82 165 133 85 43 151)(31 60 83 158 134 90 44 156)(32 53 84 163 135 95 45 149)(33 58 73 168 136 88 46 154)(34 51 74 161 137 93 47 147)(35 56 75 166 138 86 48 152)(36 49 76 159 139 91 37 145)
(1 42 170 81)(2 37 171 76)(3 44 172 83)(4 39 173 78)(5 46 174 73)(6 41 175 80)(7 48 176 75)(8 43 177 82)(9 38 178 77)(10 45 179 84)(11 40 180 79)(12 47 169 74)(13 163 188 149)(14 158 189 156)(15 165 190 151)(16 160 191 146)(17 167 192 153)(18 162 181 148)(19 157 182 155)(20 164 183 150)(21 159 184 145)(22 166 185 152)(23 161 186 147)(24 168 187 154)(25 111 140 69)(26 118 141 64)(27 113 142 71)(28 120 143 66)(29 115 144 61)(30 110 133 68)(31 117 134 63)(32 112 135 70)(33 119 136 65)(34 114 137 72)(35 109 138 67)(36 116 139 62)(49 103 91 131)(50 98 92 126)(51 105 93 121)(52 100 94 128)(53 107 95 123)(54 102 96 130)(55 97 85 125)(56 104 86 132)(57 99 87 127)(58 106 88 122)(59 101 89 129)(60 108 90 124)
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), (1,98,115,16,170,126,61,191)(2,103,116,21,171,131,62,184)(3,108,117,14,172,124,63,189)(4,101,118,19,173,129,64,182)(5,106,119,24,174,122,65,187)(6,99,120,17,175,127,66,192)(7,104,109,22,176,132,67,185)(8,97,110,15,177,125,68,190)(9,102,111,20,178,130,69,183)(10,107,112,13,179,123,70,188)(11,100,113,18,180,128,71,181)(12,105,114,23,169,121,72,186)(25,54,77,164,140,96,38,150)(26,59,78,157,141,89,39,155)(27,52,79,162,142,94,40,148)(28,57,80,167,143,87,41,153)(29,50,81,160,144,92,42,146)(30,55,82,165,133,85,43,151)(31,60,83,158,134,90,44,156)(32,53,84,163,135,95,45,149)(33,58,73,168,136,88,46,154)(34,51,74,161,137,93,47,147)(35,56,75,166,138,86,48,152)(36,49,76,159,139,91,37,145), (1,42,170,81)(2,37,171,76)(3,44,172,83)(4,39,173,78)(5,46,174,73)(6,41,175,80)(7,48,176,75)(8,43,177,82)(9,38,178,77)(10,45,179,84)(11,40,180,79)(12,47,169,74)(13,163,188,149)(14,158,189,156)(15,165,190,151)(16,160,191,146)(17,167,192,153)(18,162,181,148)(19,157,182,155)(20,164,183,150)(21,159,184,145)(22,166,185,152)(23,161,186,147)(24,168,187,154)(25,111,140,69)(26,118,141,64)(27,113,142,71)(28,120,143,66)(29,115,144,61)(30,110,133,68)(31,117,134,63)(32,112,135,70)(33,119,136,65)(34,114,137,72)(35,109,138,67)(36,116,139,62)(49,103,91,131)(50,98,92,126)(51,105,93,121)(52,100,94,128)(53,107,95,123)(54,102,96,130)(55,97,85,125)(56,104,86,132)(57,99,87,127)(58,106,88,122)(59,101,89,129)(60,108,90,124)>;
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), (1,98,115,16,170,126,61,191)(2,103,116,21,171,131,62,184)(3,108,117,14,172,124,63,189)(4,101,118,19,173,129,64,182)(5,106,119,24,174,122,65,187)(6,99,120,17,175,127,66,192)(7,104,109,22,176,132,67,185)(8,97,110,15,177,125,68,190)(9,102,111,20,178,130,69,183)(10,107,112,13,179,123,70,188)(11,100,113,18,180,128,71,181)(12,105,114,23,169,121,72,186)(25,54,77,164,140,96,38,150)(26,59,78,157,141,89,39,155)(27,52,79,162,142,94,40,148)(28,57,80,167,143,87,41,153)(29,50,81,160,144,92,42,146)(30,55,82,165,133,85,43,151)(31,60,83,158,134,90,44,156)(32,53,84,163,135,95,45,149)(33,58,73,168,136,88,46,154)(34,51,74,161,137,93,47,147)(35,56,75,166,138,86,48,152)(36,49,76,159,139,91,37,145), (1,42,170,81)(2,37,171,76)(3,44,172,83)(4,39,173,78)(5,46,174,73)(6,41,175,80)(7,48,176,75)(8,43,177,82)(9,38,178,77)(10,45,179,84)(11,40,180,79)(12,47,169,74)(13,163,188,149)(14,158,189,156)(15,165,190,151)(16,160,191,146)(17,167,192,153)(18,162,181,148)(19,157,182,155)(20,164,183,150)(21,159,184,145)(22,166,185,152)(23,161,186,147)(24,168,187,154)(25,111,140,69)(26,118,141,64)(27,113,142,71)(28,120,143,66)(29,115,144,61)(30,110,133,68)(31,117,134,63)(32,112,135,70)(33,119,136,65)(34,114,137,72)(35,109,138,67)(36,116,139,62)(49,103,91,131)(50,98,92,126)(51,105,93,121)(52,100,94,128)(53,107,95,123)(54,102,96,130)(55,97,85,125)(56,104,86,132)(57,99,87,127)(58,106,88,122)(59,101,89,129)(60,108,90,124) );
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)], [(1,98,115,16,170,126,61,191),(2,103,116,21,171,131,62,184),(3,108,117,14,172,124,63,189),(4,101,118,19,173,129,64,182),(5,106,119,24,174,122,65,187),(6,99,120,17,175,127,66,192),(7,104,109,22,176,132,67,185),(8,97,110,15,177,125,68,190),(9,102,111,20,178,130,69,183),(10,107,112,13,179,123,70,188),(11,100,113,18,180,128,71,181),(12,105,114,23,169,121,72,186),(25,54,77,164,140,96,38,150),(26,59,78,157,141,89,39,155),(27,52,79,162,142,94,40,148),(28,57,80,167,143,87,41,153),(29,50,81,160,144,92,42,146),(30,55,82,165,133,85,43,151),(31,60,83,158,134,90,44,156),(32,53,84,163,135,95,45,149),(33,58,73,168,136,88,46,154),(34,51,74,161,137,93,47,147),(35,56,75,166,138,86,48,152),(36,49,76,159,139,91,37,145)], [(1,42,170,81),(2,37,171,76),(3,44,172,83),(4,39,173,78),(5,46,174,73),(6,41,175,80),(7,48,176,75),(8,43,177,82),(9,38,178,77),(10,45,179,84),(11,40,180,79),(12,47,169,74),(13,163,188,149),(14,158,189,156),(15,165,190,151),(16,160,191,146),(17,167,192,153),(18,162,181,148),(19,157,182,155),(20,164,183,150),(21,159,184,145),(22,166,185,152),(23,161,186,147),(24,168,187,154),(25,111,140,69),(26,118,141,64),(27,113,142,71),(28,120,143,66),(29,115,144,61),(30,110,133,68),(31,117,134,63),(32,112,135,70),(33,119,136,65),(34,114,137,72),(35,109,138,67),(36,116,139,62),(49,103,91,131),(50,98,92,126),(51,105,93,121),(52,100,94,128),(53,107,95,123),(54,102,96,130),(55,97,85,125),(56,104,86,132),(57,99,87,127),(58,106,88,122),(59,101,89,129),(60,108,90,124)]])
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 |
| type | + | + | + | + | + | + | + | + | + | + | - | - | + | |
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | Q16 | C3⋊D4 | C3⋊Q16 | S3×D4 |
| kernel | C12⋊3Q16 | C4×C3⋊C8 | C12⋊2Q8 | C2×C3⋊Q16 | C3×C4⋊Q8 | C4⋊Q8 | C3⋊C8 | C2×C12 | C42 | C2×Q8 | C12 | C2×C4 | C4 | C4 |
| # reps | 1 | 1 | 1 | 4 | 1 | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 4 | 2 |
Matrix representation of C12⋊3Q16 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 56 | 12 |
| 0 | 0 | 0 | 0 | 61 | 17 |
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 22 | 32 | 0 | 0 |
| 0 | 0 | 10 | 51 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 30 | 62 | 0 | 0 | 0 | 0 |
| 62 | 43 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,56,61,0,0,0,0,12,17],[16,16,0,0,0,0,57,16,0,0,0,0,0,0,22,10,0,0,0,0,32,51,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[30,62,0,0,0,0,62,43,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C12⋊3Q16 in GAP, Magma, Sage, TeX
C_{12}\rtimes_3Q_{16} % in TeX
G:=Group("C12:3Q16"); // GroupNames label
G:=SmallGroup(192,651);
// by ID
G=gap.SmallGroup(192,651);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,120,254,219,184,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=b^4,b*a*b^-1=a^5,c*a*c^-1=a^7,c*b*c^-1=b^-1>;
// generators/relations