metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.14Q16, C4.3Dic12, C12.25SD16, C42.253D6, (C4×C8).5S3, (C4×C24).5C2, C6.1(C2×Q16), (C2×C8).281D6, (C2×C4).77D12, C6.2(C2×SD16), C4.4(C24⋊C2), (C2×C12).374D4, C12⋊2Q8.2C2, C2.4(C2×Dic12), C3⋊1(C4.SD16), C4.99(C4○D12), C6.4(C4.4D4), C22.86(C2×D12), C4⋊Dic3.2C22, C2.Dic12.1C2, C12.215(C4○D4), (C2×C12).717C23, (C2×C24).340C22, (C4×C12).303C22, C2.9(C42⋊7S3), (C2×Dic6).1C22, C2.6(C2×C24⋊C2), (C2×C6).100(C2×D4), (C2×C4).660(C22×S3), SmallGroup(192,240)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.14Q16
G = < a,b,c | a12=b8=1, c2=a6b4, ab=ba, cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 280 in 98 conjugacy classes, 47 normal (23 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C24, Dic6, C2×Dic3, C2×C12, C4×C8, Q8⋊C4, C4⋊Q8, C4⋊Dic3, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C2×Dic6, C4.SD16, C2.Dic12, C4×C24, C12⋊2Q8, C12.14Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, Q16, C2×D4, C4○D4, D12, C22×S3, C4.4D4, C2×SD16, C2×Q16, C24⋊C2, Dic12, C2×D12, C4○D12, C4.SD16, C42⋊7S3, C2×C24⋊C2, C2×Dic12, C12.14Q16
►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 35 120 21 158 100 138 151)(2 36 109 22 159 101 139 152)(3 25 110 23 160 102 140 153)(4 26 111 24 161 103 141 154)(5 27 112 13 162 104 142 155)(6 28 113 14 163 105 143 156)(7 29 114 15 164 106 144 145)(8 30 115 16 165 107 133 146)(9 31 116 17 166 108 134 147)(10 32 117 18 167 97 135 148)(11 33 118 19 168 98 136 149)(12 34 119 20 157 99 137 150)(37 191 76 178 128 53 62 90)(38 192 77 179 129 54 63 91)(39 181 78 180 130 55 64 92)(40 182 79 169 131 56 65 93)(41 183 80 170 132 57 66 94)(42 184 81 171 121 58 67 95)(43 185 82 172 122 59 68 96)(44 186 83 173 123 60 69 85)(45 187 84 174 124 49 70 86)(46 188 73 175 125 50 71 87)(47 189 74 176 126 51 72 88)(48 190 75 177 127 52 61 89)
(1 182 164 50)(2 181 165 49)(3 192 166 60)(4 191 167 59)(5 190 168 58)(6 189 157 57)(7 188 158 56)(8 187 159 55)(9 186 160 54)(10 185 161 53)(11 184 162 52)(12 183 163 51)(13 67 149 75)(14 66 150 74)(15 65 151 73)(16 64 152 84)(17 63 153 83)(18 62 154 82)(19 61 155 81)(20 72 156 80)(21 71 145 79)(22 70 146 78)(23 69 147 77)(24 68 148 76)(25 44 108 129)(26 43 97 128)(27 42 98 127)(28 41 99 126)(29 40 100 125)(30 39 101 124)(31 38 102 123)(32 37 103 122)(33 48 104 121)(34 47 105 132)(35 46 106 131)(36 45 107 130)(85 140 179 116)(86 139 180 115)(87 138 169 114)(88 137 170 113)(89 136 171 112)(90 135 172 111)(91 134 173 110)(92 133 174 109)(93 144 175 120)(94 143 176 119)(95 142 177 118)(96 141 178 117)
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), (1,35,120,21,158,100,138,151)(2,36,109,22,159,101,139,152)(3,25,110,23,160,102,140,153)(4,26,111,24,161,103,141,154)(5,27,112,13,162,104,142,155)(6,28,113,14,163,105,143,156)(7,29,114,15,164,106,144,145)(8,30,115,16,165,107,133,146)(9,31,116,17,166,108,134,147)(10,32,117,18,167,97,135,148)(11,33,118,19,168,98,136,149)(12,34,119,20,157,99,137,150)(37,191,76,178,128,53,62,90)(38,192,77,179,129,54,63,91)(39,181,78,180,130,55,64,92)(40,182,79,169,131,56,65,93)(41,183,80,170,132,57,66,94)(42,184,81,171,121,58,67,95)(43,185,82,172,122,59,68,96)(44,186,83,173,123,60,69,85)(45,187,84,174,124,49,70,86)(46,188,73,175,125,50,71,87)(47,189,74,176,126,51,72,88)(48,190,75,177,127,52,61,89), (1,182,164,50)(2,181,165,49)(3,192,166,60)(4,191,167,59)(5,190,168,58)(6,189,157,57)(7,188,158,56)(8,187,159,55)(9,186,160,54)(10,185,161,53)(11,184,162,52)(12,183,163,51)(13,67,149,75)(14,66,150,74)(15,65,151,73)(16,64,152,84)(17,63,153,83)(18,62,154,82)(19,61,155,81)(20,72,156,80)(21,71,145,79)(22,70,146,78)(23,69,147,77)(24,68,148,76)(25,44,108,129)(26,43,97,128)(27,42,98,127)(28,41,99,126)(29,40,100,125)(30,39,101,124)(31,38,102,123)(32,37,103,122)(33,48,104,121)(34,47,105,132)(35,46,106,131)(36,45,107,130)(85,140,179,116)(86,139,180,115)(87,138,169,114)(88,137,170,113)(89,136,171,112)(90,135,172,111)(91,134,173,110)(92,133,174,109)(93,144,175,120)(94,143,176,119)(95,142,177,118)(96,141,178,117)>;
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), (1,35,120,21,158,100,138,151)(2,36,109,22,159,101,139,152)(3,25,110,23,160,102,140,153)(4,26,111,24,161,103,141,154)(5,27,112,13,162,104,142,155)(6,28,113,14,163,105,143,156)(7,29,114,15,164,106,144,145)(8,30,115,16,165,107,133,146)(9,31,116,17,166,108,134,147)(10,32,117,18,167,97,135,148)(11,33,118,19,168,98,136,149)(12,34,119,20,157,99,137,150)(37,191,76,178,128,53,62,90)(38,192,77,179,129,54,63,91)(39,181,78,180,130,55,64,92)(40,182,79,169,131,56,65,93)(41,183,80,170,132,57,66,94)(42,184,81,171,121,58,67,95)(43,185,82,172,122,59,68,96)(44,186,83,173,123,60,69,85)(45,187,84,174,124,49,70,86)(46,188,73,175,125,50,71,87)(47,189,74,176,126,51,72,88)(48,190,75,177,127,52,61,89), (1,182,164,50)(2,181,165,49)(3,192,166,60)(4,191,167,59)(5,190,168,58)(6,189,157,57)(7,188,158,56)(8,187,159,55)(9,186,160,54)(10,185,161,53)(11,184,162,52)(12,183,163,51)(13,67,149,75)(14,66,150,74)(15,65,151,73)(16,64,152,84)(17,63,153,83)(18,62,154,82)(19,61,155,81)(20,72,156,80)(21,71,145,79)(22,70,146,78)(23,69,147,77)(24,68,148,76)(25,44,108,129)(26,43,97,128)(27,42,98,127)(28,41,99,126)(29,40,100,125)(30,39,101,124)(31,38,102,123)(32,37,103,122)(33,48,104,121)(34,47,105,132)(35,46,106,131)(36,45,107,130)(85,140,179,116)(86,139,180,115)(87,138,169,114)(88,137,170,113)(89,136,171,112)(90,135,172,111)(91,134,173,110)(92,133,174,109)(93,144,175,120)(94,143,176,119)(95,142,177,118)(96,141,178,117) );
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)], [(1,35,120,21,158,100,138,151),(2,36,109,22,159,101,139,152),(3,25,110,23,160,102,140,153),(4,26,111,24,161,103,141,154),(5,27,112,13,162,104,142,155),(6,28,113,14,163,105,143,156),(7,29,114,15,164,106,144,145),(8,30,115,16,165,107,133,146),(9,31,116,17,166,108,134,147),(10,32,117,18,167,97,135,148),(11,33,118,19,168,98,136,149),(12,34,119,20,157,99,137,150),(37,191,76,178,128,53,62,90),(38,192,77,179,129,54,63,91),(39,181,78,180,130,55,64,92),(40,182,79,169,131,56,65,93),(41,183,80,170,132,57,66,94),(42,184,81,171,121,58,67,95),(43,185,82,172,122,59,68,96),(44,186,83,173,123,60,69,85),(45,187,84,174,124,49,70,86),(46,188,73,175,125,50,71,87),(47,189,74,176,126,51,72,88),(48,190,75,177,127,52,61,89)], [(1,182,164,50),(2,181,165,49),(3,192,166,60),(4,191,167,59),(5,190,168,58),(6,189,157,57),(7,188,158,56),(8,187,159,55),(9,186,160,54),(10,185,161,53),(11,184,162,52),(12,183,163,51),(13,67,149,75),(14,66,150,74),(15,65,151,73),(16,64,152,84),(17,63,153,83),(18,62,154,82),(19,61,155,81),(20,72,156,80),(21,71,145,79),(22,70,146,78),(23,69,147,77),(24,68,148,76),(25,44,108,129),(26,43,97,128),(27,42,98,127),(28,41,99,126),(29,40,100,125),(30,39,101,124),(31,38,102,123),(32,37,103,122),(33,48,104,121),(34,47,105,132),(35,46,106,131),(36,45,107,130),(85,140,179,116),(86,139,180,115),(87,138,169,114),(88,137,170,113),(89,136,171,112),(90,135,172,111),(91,134,173,110),(92,133,174,109),(93,144,175,120),(94,143,176,119),(95,142,177,118),(96,141,178,117)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | ··· | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | ··· | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | ··· | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C2 | S3 | D4 | D6 | D6 | SD16 | Q16 | C4○D4 | D12 | C24⋊C2 | Dic12 | C4○D12 |
| kernel | C12.14Q16 | C2.Dic12 | C4×C24 | C12⋊2Q8 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C12 | C12 | C2×C4 | C4 | C4 | C4 |
| # reps | 1 | 4 | 1 | 2 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C12.14Q16 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 14 | 7 |
| 0 | 0 | 66 | 7 |
| 25 | 62 | 0 | 0 |
| 11 | 36 | 0 | 0 |
| 0 | 0 | 43 | 13 |
| 0 | 0 | 60 | 30 |
| 54 | 29 | 0 | 0 |
| 48 | 19 | 0 | 0 |
| 0 | 0 | 53 | 55 |
| 0 | 0 | 2 | 20 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,14,66,0,0,7,7],[25,11,0,0,62,36,0,0,0,0,43,60,0,0,13,30],[54,48,0,0,29,19,0,0,0,0,53,2,0,0,55,20] >;
C12.14Q16 in GAP, Magma, Sage, TeX
C_{12}._{14}Q_{16} % in TeX
G:=Group("C12.14Q16"); // GroupNames label
G:=SmallGroup(192,240);
// by ID
G=gap.SmallGroup(192,240);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,142,1123,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6*b^4,a*b=b*a,c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations