metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.1Q16, C42.3D6, C4.5Dic12, C12.35SD16, C4⋊C8.3S3, C12⋊C8.8C2, C4.9(C24⋊C2), (C2×C12).463D4, (C2×C4).121D12, C4.9(D4.S3), C12⋊2Q8.7C2, (C2×Dic6).2C4, C3⋊1(C4.6Q16), C4.9(C3⋊Q16), (C4×C12).39C22, C6.7(Q8⋊C4), C6.3(C4.D4), C22.60(D6⋊C4), C2.4(C6.SD16), C2.4(C2.Dic12), C2.4(C12.46D4), (C3×C4⋊C8).3C2, (C2×C4).14(C4×S3), (C2×C12).26(C2×C4), (C2×C4).227(C3⋊D4), (C2×C6).43(C22⋊C4), SmallGroup(192,40)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.Dic12
G = < a,b,c | a4=b24=1, c2=ab12, bab-1=a-1, ac=ca, cbc-1=a-1b-1 >
Subgroups: 184 in 64 conjugacy classes, 33 normal (31 characteristic)
C1, C2, C3, C4, C4, C22, C6, C8, C2×C4, C2×C4, Q8, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×C12, C4⋊C8, C4⋊C8, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4.6Q16, C12⋊C8, C3×C4⋊C8, C12⋊2Q8, C4.Dic12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, SD16, Q16, C4×S3, D12, C3⋊D4, C4.D4, Q8⋊C4, C24⋊C2, Dic12, D6⋊C4, D4.S3, C3⋊Q16, C4.6Q16, C6.SD16, C2.Dic12, C12.46D4, C4.Dic12
►Regular action on 192 points
Generators in S192
(1 41 175 164)(2 165 176 42)(3 43 177 166)(4 167 178 44)(5 45 179 168)(6 145 180 46)(7 47 181 146)(8 147 182 48)(9 25 183 148)(10 149 184 26)(11 27 185 150)(12 151 186 28)(13 29 187 152)(14 153 188 30)(15 31 189 154)(16 155 190 32)(17 33 191 156)(18 157 192 34)(19 35 169 158)(20 159 170 36)(21 37 171 160)(22 161 172 38)(23 39 173 162)(24 163 174 40)(49 77 102 123)(50 124 103 78)(51 79 104 125)(52 126 105 80)(53 81 106 127)(54 128 107 82)(55 83 108 129)(56 130 109 84)(57 85 110 131)(58 132 111 86)(59 87 112 133)(60 134 113 88)(61 89 114 135)(62 136 115 90)(63 91 116 137)(64 138 117 92)(65 93 118 139)(66 140 119 94)(67 95 120 141)(68 142 97 96)(69 73 98 143)(70 144 99 74)(71 75 100 121)(72 122 101 76)
(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 93 29 106 175 139 152 53)(2 117 153 80 176 64 30 126)(3 91 31 104 177 137 154 51)(4 115 155 78 178 62 32 124)(5 89 33 102 179 135 156 49)(6 113 157 76 180 60 34 122)(7 87 35 100 181 133 158 71)(8 111 159 74 182 58 36 144)(9 85 37 98 183 131 160 69)(10 109 161 96 184 56 38 142)(11 83 39 120 185 129 162 67)(12 107 163 94 186 54 40 140)(13 81 41 118 187 127 164 65)(14 105 165 92 188 52 42 138)(15 79 43 116 189 125 166 63)(16 103 167 90 190 50 44 136)(17 77 45 114 191 123 168 61)(18 101 145 88 192 72 46 134)(19 75 47 112 169 121 146 59)(20 99 147 86 170 70 48 132)(21 73 25 110 171 143 148 57)(22 97 149 84 172 68 26 130)(23 95 27 108 173 141 150 55)(24 119 151 82 174 66 28 128)
G:=sub<Sym(192)| (1,41,175,164)(2,165,176,42)(3,43,177,166)(4,167,178,44)(5,45,179,168)(6,145,180,46)(7,47,181,146)(8,147,182,48)(9,25,183,148)(10,149,184,26)(11,27,185,150)(12,151,186,28)(13,29,187,152)(14,153,188,30)(15,31,189,154)(16,155,190,32)(17,33,191,156)(18,157,192,34)(19,35,169,158)(20,159,170,36)(21,37,171,160)(22,161,172,38)(23,39,173,162)(24,163,174,40)(49,77,102,123)(50,124,103,78)(51,79,104,125)(52,126,105,80)(53,81,106,127)(54,128,107,82)(55,83,108,129)(56,130,109,84)(57,85,110,131)(58,132,111,86)(59,87,112,133)(60,134,113,88)(61,89,114,135)(62,136,115,90)(63,91,116,137)(64,138,117,92)(65,93,118,139)(66,140,119,94)(67,95,120,141)(68,142,97,96)(69,73,98,143)(70,144,99,74)(71,75,100,121)(72,122,101,76), (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), (1,93,29,106,175,139,152,53)(2,117,153,80,176,64,30,126)(3,91,31,104,177,137,154,51)(4,115,155,78,178,62,32,124)(5,89,33,102,179,135,156,49)(6,113,157,76,180,60,34,122)(7,87,35,100,181,133,158,71)(8,111,159,74,182,58,36,144)(9,85,37,98,183,131,160,69)(10,109,161,96,184,56,38,142)(11,83,39,120,185,129,162,67)(12,107,163,94,186,54,40,140)(13,81,41,118,187,127,164,65)(14,105,165,92,188,52,42,138)(15,79,43,116,189,125,166,63)(16,103,167,90,190,50,44,136)(17,77,45,114,191,123,168,61)(18,101,145,88,192,72,46,134)(19,75,47,112,169,121,146,59)(20,99,147,86,170,70,48,132)(21,73,25,110,171,143,148,57)(22,97,149,84,172,68,26,130)(23,95,27,108,173,141,150,55)(24,119,151,82,174,66,28,128)>;
G:=Group( (1,41,175,164)(2,165,176,42)(3,43,177,166)(4,167,178,44)(5,45,179,168)(6,145,180,46)(7,47,181,146)(8,147,182,48)(9,25,183,148)(10,149,184,26)(11,27,185,150)(12,151,186,28)(13,29,187,152)(14,153,188,30)(15,31,189,154)(16,155,190,32)(17,33,191,156)(18,157,192,34)(19,35,169,158)(20,159,170,36)(21,37,171,160)(22,161,172,38)(23,39,173,162)(24,163,174,40)(49,77,102,123)(50,124,103,78)(51,79,104,125)(52,126,105,80)(53,81,106,127)(54,128,107,82)(55,83,108,129)(56,130,109,84)(57,85,110,131)(58,132,111,86)(59,87,112,133)(60,134,113,88)(61,89,114,135)(62,136,115,90)(63,91,116,137)(64,138,117,92)(65,93,118,139)(66,140,119,94)(67,95,120,141)(68,142,97,96)(69,73,98,143)(70,144,99,74)(71,75,100,121)(72,122,101,76), (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), (1,93,29,106,175,139,152,53)(2,117,153,80,176,64,30,126)(3,91,31,104,177,137,154,51)(4,115,155,78,178,62,32,124)(5,89,33,102,179,135,156,49)(6,113,157,76,180,60,34,122)(7,87,35,100,181,133,158,71)(8,111,159,74,182,58,36,144)(9,85,37,98,183,131,160,69)(10,109,161,96,184,56,38,142)(11,83,39,120,185,129,162,67)(12,107,163,94,186,54,40,140)(13,81,41,118,187,127,164,65)(14,105,165,92,188,52,42,138)(15,79,43,116,189,125,166,63)(16,103,167,90,190,50,44,136)(17,77,45,114,191,123,168,61)(18,101,145,88,192,72,46,134)(19,75,47,112,169,121,146,59)(20,99,147,86,170,70,48,132)(21,73,25,110,171,143,148,57)(22,97,149,84,172,68,26,130)(23,95,27,108,173,141,150,55)(24,119,151,82,174,66,28,128) );
G=PermutationGroup([[(1,41,175,164),(2,165,176,42),(3,43,177,166),(4,167,178,44),(5,45,179,168),(6,145,180,46),(7,47,181,146),(8,147,182,48),(9,25,183,148),(10,149,184,26),(11,27,185,150),(12,151,186,28),(13,29,187,152),(14,153,188,30),(15,31,189,154),(16,155,190,32),(17,33,191,156),(18,157,192,34),(19,35,169,158),(20,159,170,36),(21,37,171,160),(22,161,172,38),(23,39,173,162),(24,163,174,40),(49,77,102,123),(50,124,103,78),(51,79,104,125),(52,126,105,80),(53,81,106,127),(54,128,107,82),(55,83,108,129),(56,130,109,84),(57,85,110,131),(58,132,111,86),(59,87,112,133),(60,134,113,88),(61,89,114,135),(62,136,115,90),(63,91,116,137),(64,138,117,92),(65,93,118,139),(66,140,119,94),(67,95,120,141),(68,142,97,96),(69,73,98,143),(70,144,99,74),(71,75,100,121),(72,122,101,76)], [(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)], [(1,93,29,106,175,139,152,53),(2,117,153,80,176,64,30,126),(3,91,31,104,177,137,154,51),(4,115,155,78,178,62,32,124),(5,89,33,102,179,135,156,49),(6,113,157,76,180,60,34,122),(7,87,35,100,181,133,158,71),(8,111,159,74,182,58,36,144),(9,85,37,98,183,131,160,69),(10,109,161,96,184,56,38,142),(11,83,39,120,185,129,162,67),(12,107,163,94,186,54,40,140),(13,81,41,118,187,127,164,65),(14,105,165,92,188,52,42,138),(15,79,43,116,189,125,166,63),(16,103,167,90,190,50,44,136),(17,77,45,114,191,123,168,61),(18,101,145,88,192,72,46,134),(19,75,47,112,169,121,146,59),(20,99,147,86,170,70,48,132),(21,73,25,110,171,143,148,57),(22,97,149,84,172,68,26,130),(23,95,27,108,173,141,150,55),(24,119,151,82,174,66,28,128)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | + | - | + | - | - | + | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C24⋊C2 | Dic12 | C4.D4 | D4.S3 | C3⋊Q16 | C12.46D4 |
| kernel | C4.Dic12 | C12⋊C8 | C3×C4⋊C8 | C12⋊2Q8 | C2×Dic6 | C4⋊C8 | C2×C12 | C42 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 |
Matrix representation of C4.Dic12 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 71 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 0 | 57 | 0 | 0 | 0 | 0 |
| 32 | 41 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 43 | 0 | 0 |
| 0 | 0 | 30 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 68 | 54 |
| 0 | 0 | 0 | 0 | 32 | 5 |
| 58 | 32 | 0 | 0 | 0 | 0 |
| 34 | 15 | 0 | 0 | 0 | 0 |
| 0 | 0 | 10 | 51 | 0 | 0 |
| 0 | 0 | 61 | 63 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 67 | 0 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,71,1],[0,32,0,0,0,0,57,41,0,0,0,0,0,0,43,30,0,0,0,0,43,13,0,0,0,0,0,0,68,32,0,0,0,0,54,5],[58,34,0,0,0,0,32,15,0,0,0,0,0,0,10,61,0,0,0,0,51,63,0,0,0,0,0,0,12,67,0,0,0,0,12,0] >;
C4.Dic12 in GAP, Magma, Sage, TeX
C_4.{\rm Dic}_{12} % in TeX
G:=Group("C4.Dic12"); // GroupNames label
G:=SmallGroup(192,40);
// by ID
G=gap.SmallGroup(192,40);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,85,92,422,387,268,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=1,c^2=a*b^12,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=a^-1*b^-1>;
// generators/relations