metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.47D8, C12.2Q16, C42.4D6, C4.6Dic12, C12.46SD16, C4⋊C8.4S3, C4⋊Dic3.2C4, C12⋊C8.9C2, C4.19(D4⋊S3), (C2×C12).464D4, (C2×C4).122D12, C3⋊1(C4.10D8), C12⋊2Q8.8C2, C4.10(C24⋊C2), C6.2(D4⋊C4), (C4×C12).40C22, C6.8(Q8⋊C4), C2.4(C6.D8), C22.61(D6⋊C4), C4.11(Q8⋊2S3), C6.3(C4.10D4), C2.5(C2.Dic12), C2.4(C12.47D4), (C3×C4⋊C8).4C2, (C2×C4).15(C4×S3), (C2×C12).27(C2×C4), (C2×C4).228(C3⋊D4), (C2×C6).44(C22⋊C4), SmallGroup(192,41)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.47D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a9b-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⋊Dic3, C4×C12, C2×C24, C2×Dic6, C4.10D8, C12⋊C8, C3×C4⋊C8, C12⋊2Q8, C12.47D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, D6, C22⋊C4, D8, SD16, Q16, C4×S3, D12, C3⋊D4, C4.10D4, D4⋊C4, Q8⋊C4, C24⋊C2, Dic12, D6⋊C4, D4⋊S3, Q8⋊2S3, C4.10D8, C6.D8, C2.Dic12, C12.47D4, C12.47D8
►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 38 114 156 32 131 17 136)(2 37 115 155 33 130 18 135)(3 48 116 154 34 129 19 134)(4 47 117 153 35 128 20 133)(5 46 118 152 36 127 21 144)(6 45 119 151 25 126 22 143)(7 44 120 150 26 125 23 142)(8 43 109 149 27 124 24 141)(9 42 110 148 28 123 13 140)(10 41 111 147 29 122 14 139)(11 40 112 146 30 121 15 138)(12 39 113 145 31 132 16 137)(49 169 158 97 72 82 86 186)(50 180 159 108 61 81 87 185)(51 179 160 107 62 80 88 184)(52 178 161 106 63 79 89 183)(53 177 162 105 64 78 90 182)(54 176 163 104 65 77 91 181)(55 175 164 103 66 76 92 192)(56 174 165 102 67 75 93 191)(57 173 166 101 68 74 94 190)(58 172 167 100 69 73 95 189)(59 171 168 99 70 84 96 188)(60 170 157 98 71 83 85 187)
(1 169 7 175)(2 180 8 174)(3 179 9 173)(4 178 10 172)(5 177 11 171)(6 176 12 170)(13 101 19 107)(14 100 20 106)(15 99 21 105)(16 98 22 104)(17 97 23 103)(18 108 24 102)(25 77 31 83)(26 76 32 82)(27 75 33 81)(28 74 34 80)(29 73 35 79)(30 84 36 78)(37 53 43 59)(38 52 44 58)(39 51 45 57)(40 50 46 56)(41 49 47 55)(42 60 48 54)(61 127 67 121)(62 126 68 132)(63 125 69 131)(64 124 70 130)(65 123 71 129)(66 122 72 128)(85 154 91 148)(86 153 92 147)(87 152 93 146)(88 151 94 145)(89 150 95 156)(90 149 96 155)(109 191 115 185)(110 190 116 184)(111 189 117 183)(112 188 118 182)(113 187 119 181)(114 186 120 192)(133 164 139 158)(134 163 140 157)(135 162 141 168)(136 161 142 167)(137 160 143 166)(138 159 144 165)
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), (1,38,114,156,32,131,17,136)(2,37,115,155,33,130,18,135)(3,48,116,154,34,129,19,134)(4,47,117,153,35,128,20,133)(5,46,118,152,36,127,21,144)(6,45,119,151,25,126,22,143)(7,44,120,150,26,125,23,142)(8,43,109,149,27,124,24,141)(9,42,110,148,28,123,13,140)(10,41,111,147,29,122,14,139)(11,40,112,146,30,121,15,138)(12,39,113,145,31,132,16,137)(49,169,158,97,72,82,86,186)(50,180,159,108,61,81,87,185)(51,179,160,107,62,80,88,184)(52,178,161,106,63,79,89,183)(53,177,162,105,64,78,90,182)(54,176,163,104,65,77,91,181)(55,175,164,103,66,76,92,192)(56,174,165,102,67,75,93,191)(57,173,166,101,68,74,94,190)(58,172,167,100,69,73,95,189)(59,171,168,99,70,84,96,188)(60,170,157,98,71,83,85,187), (1,169,7,175)(2,180,8,174)(3,179,9,173)(4,178,10,172)(5,177,11,171)(6,176,12,170)(13,101,19,107)(14,100,20,106)(15,99,21,105)(16,98,22,104)(17,97,23,103)(18,108,24,102)(25,77,31,83)(26,76,32,82)(27,75,33,81)(28,74,34,80)(29,73,35,79)(30,84,36,78)(37,53,43,59)(38,52,44,58)(39,51,45,57)(40,50,46,56)(41,49,47,55)(42,60,48,54)(61,127,67,121)(62,126,68,132)(63,125,69,131)(64,124,70,130)(65,123,71,129)(66,122,72,128)(85,154,91,148)(86,153,92,147)(87,152,93,146)(88,151,94,145)(89,150,95,156)(90,149,96,155)(109,191,115,185)(110,190,116,184)(111,189,117,183)(112,188,118,182)(113,187,119,181)(114,186,120,192)(133,164,139,158)(134,163,140,157)(135,162,141,168)(136,161,142,167)(137,160,143,166)(138,159,144,165)>;
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), (1,38,114,156,32,131,17,136)(2,37,115,155,33,130,18,135)(3,48,116,154,34,129,19,134)(4,47,117,153,35,128,20,133)(5,46,118,152,36,127,21,144)(6,45,119,151,25,126,22,143)(7,44,120,150,26,125,23,142)(8,43,109,149,27,124,24,141)(9,42,110,148,28,123,13,140)(10,41,111,147,29,122,14,139)(11,40,112,146,30,121,15,138)(12,39,113,145,31,132,16,137)(49,169,158,97,72,82,86,186)(50,180,159,108,61,81,87,185)(51,179,160,107,62,80,88,184)(52,178,161,106,63,79,89,183)(53,177,162,105,64,78,90,182)(54,176,163,104,65,77,91,181)(55,175,164,103,66,76,92,192)(56,174,165,102,67,75,93,191)(57,173,166,101,68,74,94,190)(58,172,167,100,69,73,95,189)(59,171,168,99,70,84,96,188)(60,170,157,98,71,83,85,187), (1,169,7,175)(2,180,8,174)(3,179,9,173)(4,178,10,172)(5,177,11,171)(6,176,12,170)(13,101,19,107)(14,100,20,106)(15,99,21,105)(16,98,22,104)(17,97,23,103)(18,108,24,102)(25,77,31,83)(26,76,32,82)(27,75,33,81)(28,74,34,80)(29,73,35,79)(30,84,36,78)(37,53,43,59)(38,52,44,58)(39,51,45,57)(40,50,46,56)(41,49,47,55)(42,60,48,54)(61,127,67,121)(62,126,68,132)(63,125,69,131)(64,124,70,130)(65,123,71,129)(66,122,72,128)(85,154,91,148)(86,153,92,147)(87,152,93,146)(88,151,94,145)(89,150,95,156)(90,149,96,155)(109,191,115,185)(110,190,116,184)(111,189,117,183)(112,188,118,182)(113,187,119,181)(114,186,120,192)(133,164,139,158)(134,163,140,157)(135,162,141,168)(136,161,142,167)(137,160,143,166)(138,159,144,165) );
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)], [(1,38,114,156,32,131,17,136),(2,37,115,155,33,130,18,135),(3,48,116,154,34,129,19,134),(4,47,117,153,35,128,20,133),(5,46,118,152,36,127,21,144),(6,45,119,151,25,126,22,143),(7,44,120,150,26,125,23,142),(8,43,109,149,27,124,24,141),(9,42,110,148,28,123,13,140),(10,41,111,147,29,122,14,139),(11,40,112,146,30,121,15,138),(12,39,113,145,31,132,16,137),(49,169,158,97,72,82,86,186),(50,180,159,108,61,81,87,185),(51,179,160,107,62,80,88,184),(52,178,161,106,63,79,89,183),(53,177,162,105,64,78,90,182),(54,176,163,104,65,77,91,181),(55,175,164,103,66,76,92,192),(56,174,165,102,67,75,93,191),(57,173,166,101,68,74,94,190),(58,172,167,100,69,73,95,189),(59,171,168,99,70,84,96,188),(60,170,157,98,71,83,85,187)], [(1,169,7,175),(2,180,8,174),(3,179,9,173),(4,178,10,172),(5,177,11,171),(6,176,12,170),(13,101,19,107),(14,100,20,106),(15,99,21,105),(16,98,22,104),(17,97,23,103),(18,108,24,102),(25,77,31,83),(26,76,32,82),(27,75,33,81),(28,74,34,80),(29,73,35,79),(30,84,36,78),(37,53,43,59),(38,52,44,58),(39,51,45,57),(40,50,46,56),(41,49,47,55),(42,60,48,54),(61,127,67,121),(62,126,68,132),(63,125,69,131),(64,124,70,130),(65,123,71,129),(66,122,72,128),(85,154,91,148),(86,153,92,147),(87,152,93,146),(88,151,94,145),(89,150,95,156),(90,149,96,155),(109,191,115,185),(110,190,116,184),(111,189,117,183),(112,188,118,182),(113,187,119,181),(114,186,120,192),(133,164,139,158),(134,163,140,157),(135,162,141,168),(136,161,142,167),(137,160,143,166),(138,159,144,165)]])
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 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | - | - | + | + | - | |||||
| image | C1 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D8 | SD16 | Q16 | C4×S3 | D12 | C3⋊D4 | C24⋊C2 | Dic12 | C4.10D4 | D4⋊S3 | Q8⋊2S3 | C12.47D4 |
| kernel | C12.47D8 | C12⋊C8 | C3×C4⋊C8 | C12⋊2Q8 | C4⋊Dic3 | C4⋊C8 | C2×C12 | C42 | C12 | C12 | C12 | C2×C4 | C2×C4 | C2×C4 | C4 | C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 |
Matrix representation of C12.47D8 ►in GL4(𝔽73) generated by
| 59 | 66 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 10 | 51 | 0 | 0 |
| 41 | 63 | 0 | 0 |
| 0 | 0 | 0 | 38 |
| 0 | 0 | 48 | 32 |
| 8 | 47 | 0 | 0 |
| 39 | 65 | 0 | 0 |
| 0 | 0 | 71 | 52 |
| 0 | 0 | 21 | 2 |
G:=sub<GL(4,GF(73))| [59,7,0,0,66,66,0,0,0,0,1,0,0,0,0,1],[10,41,0,0,51,63,0,0,0,0,0,48,0,0,38,32],[8,39,0,0,47,65,0,0,0,0,71,21,0,0,52,2] >;
C12.47D8 in GAP, Magma, Sage, TeX
C_{12}._{47}D_8 % in TeX
G:=Group("C12.47D8"); // GroupNames label
G:=SmallGroup(192,41);
// by ID
G=gap.SmallGroup(192,41);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,56,85,316,422,387,268,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^9*b^-1>;
// generators/relations