metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.26D4, Dic3⋊1Q16, C3⋊C8.21D4, C4.27(S3×D4), C3⋊3(C4⋊Q16), C12.51(C2×D4), (C2×C8).242D6, (C2×Q16).4S3, (C6×Q16).5C2, (C2×Q8).84D6, C6.28(C2×Q16), C2.17(S3×Q16), C8.18(C3⋊D4), (C8×Dic3).5C2, C6.33(C4⋊1D4), (C2×C24).94C22, C22.275(S3×D4), (C6×Q8).86C22, C2.24(C12⋊3D4), (C2×C12).457C23, Dic3⋊Q8.6C2, (C2×Dic3).116D4, (C2×Dic12).11C2, (C2×Dic6).131C22, (C4×Dic3).244C22, C4.14(C2×C3⋊D4), (C2×C6).368(C2×D4), (C2×C3⋊Q16).9C2, (C2×C3⋊C8).277C22, (C2×C4).545(C22×S3), SmallGroup(192,742)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.26D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a-1, cbc-1=b-1 >
Subgroups: 312 in 122 conjugacy classes, 47 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C8, C2×C4, C2×C4, Q8, Dic3, Dic3, C12, C12, C2×C6, C42, C4⋊C4, C2×C8, C2×C8, Q16, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C4×C8, C4⋊Q8, C2×Q16, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C3⋊Q16, C2×C24, C3×Q16, C2×Dic6, C6×Q8, C4⋊Q16, C8×Dic3, C2×Dic12, C2×C3⋊Q16, Dic3⋊Q8, C6×Q16, C24.26D4
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C2×Q16, S3×D4, C2×C3⋊D4, C4⋊Q16, S3×Q16, C12⋊3D4, C24.26D4
►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)
(1 148 78 107)(2 165 79 100)(3 158 80 117)(4 151 81 110)(5 168 82 103)(6 161 83 120)(7 154 84 113)(8 147 85 106)(9 164 86 99)(10 157 87 116)(11 150 88 109)(12 167 89 102)(13 160 90 119)(14 153 91 112)(15 146 92 105)(16 163 93 98)(17 156 94 115)(18 149 95 108)(19 166 96 101)(20 159 73 118)(21 152 74 111)(22 145 75 104)(23 162 76 97)(24 155 77 114)(25 123 173 49)(26 140 174 66)(27 133 175 59)(28 126 176 52)(29 143 177 69)(30 136 178 62)(31 129 179 55)(32 122 180 72)(33 139 181 65)(34 132 182 58)(35 125 183 51)(36 142 184 68)(37 135 185 61)(38 128 186 54)(39 121 187 71)(40 138 188 64)(41 131 189 57)(42 124 190 50)(43 141 191 67)(44 134 192 60)(45 127 169 53)(46 144 170 70)(47 137 171 63)(48 130 172 56)
(1 135 13 123)(2 134 14 122)(3 133 15 121)(4 132 16 144)(5 131 17 143)(6 130 18 142)(7 129 19 141)(8 128 20 140)(9 127 21 139)(10 126 22 138)(11 125 23 137)(12 124 24 136)(25 148 37 160)(26 147 38 159)(27 146 39 158)(28 145 40 157)(29 168 41 156)(30 167 42 155)(31 166 43 154)(32 165 44 153)(33 164 45 152)(34 163 46 151)(35 162 47 150)(36 161 48 149)(49 78 61 90)(50 77 62 89)(51 76 63 88)(52 75 64 87)(53 74 65 86)(54 73 66 85)(55 96 67 84)(56 95 68 83)(57 94 69 82)(58 93 70 81)(59 92 71 80)(60 91 72 79)(97 171 109 183)(98 170 110 182)(99 169 111 181)(100 192 112 180)(101 191 113 179)(102 190 114 178)(103 189 115 177)(104 188 116 176)(105 187 117 175)(106 186 118 174)(107 185 119 173)(108 184 120 172)
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), (1,148,78,107)(2,165,79,100)(3,158,80,117)(4,151,81,110)(5,168,82,103)(6,161,83,120)(7,154,84,113)(8,147,85,106)(9,164,86,99)(10,157,87,116)(11,150,88,109)(12,167,89,102)(13,160,90,119)(14,153,91,112)(15,146,92,105)(16,163,93,98)(17,156,94,115)(18,149,95,108)(19,166,96,101)(20,159,73,118)(21,152,74,111)(22,145,75,104)(23,162,76,97)(24,155,77,114)(25,123,173,49)(26,140,174,66)(27,133,175,59)(28,126,176,52)(29,143,177,69)(30,136,178,62)(31,129,179,55)(32,122,180,72)(33,139,181,65)(34,132,182,58)(35,125,183,51)(36,142,184,68)(37,135,185,61)(38,128,186,54)(39,121,187,71)(40,138,188,64)(41,131,189,57)(42,124,190,50)(43,141,191,67)(44,134,192,60)(45,127,169,53)(46,144,170,70)(47,137,171,63)(48,130,172,56), (1,135,13,123)(2,134,14,122)(3,133,15,121)(4,132,16,144)(5,131,17,143)(6,130,18,142)(7,129,19,141)(8,128,20,140)(9,127,21,139)(10,126,22,138)(11,125,23,137)(12,124,24,136)(25,148,37,160)(26,147,38,159)(27,146,39,158)(28,145,40,157)(29,168,41,156)(30,167,42,155)(31,166,43,154)(32,165,44,153)(33,164,45,152)(34,163,46,151)(35,162,47,150)(36,161,48,149)(49,78,61,90)(50,77,62,89)(51,76,63,88)(52,75,64,87)(53,74,65,86)(54,73,66,85)(55,96,67,84)(56,95,68,83)(57,94,69,82)(58,93,70,81)(59,92,71,80)(60,91,72,79)(97,171,109,183)(98,170,110,182)(99,169,111,181)(100,192,112,180)(101,191,113,179)(102,190,114,178)(103,189,115,177)(104,188,116,176)(105,187,117,175)(106,186,118,174)(107,185,119,173)(108,184,120,172)>;
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), (1,148,78,107)(2,165,79,100)(3,158,80,117)(4,151,81,110)(5,168,82,103)(6,161,83,120)(7,154,84,113)(8,147,85,106)(9,164,86,99)(10,157,87,116)(11,150,88,109)(12,167,89,102)(13,160,90,119)(14,153,91,112)(15,146,92,105)(16,163,93,98)(17,156,94,115)(18,149,95,108)(19,166,96,101)(20,159,73,118)(21,152,74,111)(22,145,75,104)(23,162,76,97)(24,155,77,114)(25,123,173,49)(26,140,174,66)(27,133,175,59)(28,126,176,52)(29,143,177,69)(30,136,178,62)(31,129,179,55)(32,122,180,72)(33,139,181,65)(34,132,182,58)(35,125,183,51)(36,142,184,68)(37,135,185,61)(38,128,186,54)(39,121,187,71)(40,138,188,64)(41,131,189,57)(42,124,190,50)(43,141,191,67)(44,134,192,60)(45,127,169,53)(46,144,170,70)(47,137,171,63)(48,130,172,56), (1,135,13,123)(2,134,14,122)(3,133,15,121)(4,132,16,144)(5,131,17,143)(6,130,18,142)(7,129,19,141)(8,128,20,140)(9,127,21,139)(10,126,22,138)(11,125,23,137)(12,124,24,136)(25,148,37,160)(26,147,38,159)(27,146,39,158)(28,145,40,157)(29,168,41,156)(30,167,42,155)(31,166,43,154)(32,165,44,153)(33,164,45,152)(34,163,46,151)(35,162,47,150)(36,161,48,149)(49,78,61,90)(50,77,62,89)(51,76,63,88)(52,75,64,87)(53,74,65,86)(54,73,66,85)(55,96,67,84)(56,95,68,83)(57,94,69,82)(58,93,70,81)(59,92,71,80)(60,91,72,79)(97,171,109,183)(98,170,110,182)(99,169,111,181)(100,192,112,180)(101,191,113,179)(102,190,114,178)(103,189,115,177)(104,188,116,176)(105,187,117,175)(106,186,118,174)(107,185,119,173)(108,184,120,172) );
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)], [(1,148,78,107),(2,165,79,100),(3,158,80,117),(4,151,81,110),(5,168,82,103),(6,161,83,120),(7,154,84,113),(8,147,85,106),(9,164,86,99),(10,157,87,116),(11,150,88,109),(12,167,89,102),(13,160,90,119),(14,153,91,112),(15,146,92,105),(16,163,93,98),(17,156,94,115),(18,149,95,108),(19,166,96,101),(20,159,73,118),(21,152,74,111),(22,145,75,104),(23,162,76,97),(24,155,77,114),(25,123,173,49),(26,140,174,66),(27,133,175,59),(28,126,176,52),(29,143,177,69),(30,136,178,62),(31,129,179,55),(32,122,180,72),(33,139,181,65),(34,132,182,58),(35,125,183,51),(36,142,184,68),(37,135,185,61),(38,128,186,54),(39,121,187,71),(40,138,188,64),(41,131,189,57),(42,124,190,50),(43,141,191,67),(44,134,192,60),(45,127,169,53),(46,144,170,70),(47,137,171,63),(48,130,172,56)], [(1,135,13,123),(2,134,14,122),(3,133,15,121),(4,132,16,144),(5,131,17,143),(6,130,18,142),(7,129,19,141),(8,128,20,140),(9,127,21,139),(10,126,22,138),(11,125,23,137),(12,124,24,136),(25,148,37,160),(26,147,38,159),(27,146,39,158),(28,145,40,157),(29,168,41,156),(30,167,42,155),(31,166,43,154),(32,165,44,153),(33,164,45,152),(34,163,46,151),(35,162,47,150),(36,161,48,149),(49,78,61,90),(50,77,62,89),(51,76,63,88),(52,75,64,87),(53,74,65,86),(54,73,66,85),(55,96,67,84),(56,95,68,83),(57,94,69,82),(58,93,70,81),(59,92,71,80),(60,91,72,79),(97,171,109,183),(98,170,110,182),(99,169,111,181),(100,192,112,180),(101,191,113,179),(102,190,114,178),(103,189,115,177),(104,188,116,176),(105,187,117,175),(106,186,118,174),(107,185,119,173),(108,184,120,172)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | Q16 | C3⋊D4 | S3×D4 | S3×D4 | S3×Q16 |
| kernel | C24.26D4 | C8×Dic3 | C2×Dic12 | C2×C3⋊Q16 | Dic3⋊Q8 | C6×Q16 | C2×Q16 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×Q8 | Dic3 | C8 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 8 | 4 | 1 | 1 | 4 |
Matrix representation of C24.26D4 ►in GL6(𝔽73)
| 16 | 57 | 0 | 0 | 0 | 0 |
| 16 | 16 | 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 | 1 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 69 | 0 | 0 |
| 0 | 0 | 17 | 33 | 0 | 0 |
| 0 | 0 | 0 | 0 | 11 | 2 |
| 0 | 0 | 0 | 0 | 13 | 62 |
| 43 | 62 | 0 | 0 | 0 | 0 |
| 62 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 31 | 0 | 0 |
| 0 | 0 | 72 | 55 | 0 | 0 |
| 0 | 0 | 0 | 0 | 62 | 71 |
| 0 | 0 | 0 | 0 | 60 | 11 |
G:=sub<GL(6,GF(73))| [16,16,0,0,0,0,57,16,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,1],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,40,17,0,0,0,0,69,33,0,0,0,0,0,0,11,13,0,0,0,0,2,62],[43,62,0,0,0,0,62,30,0,0,0,0,0,0,18,72,0,0,0,0,31,55,0,0,0,0,0,0,62,60,0,0,0,0,71,11] >;
C24.26D4 in GAP, Magma, Sage, TeX
C_{24}._{26}D_4 % in TeX
G:=Group("C24.26D4"); // GroupNames label
G:=SmallGroup(192,742);
// by ID
G=gap.SmallGroup(192,742);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,232,422,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=b^-1>;
// generators/relations