metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊5C4, C2.1D48, C6.4D16, C6.2Q32, C16⋊3Dic3, C24.11Q8, C12.12Q16, C8.10Dic6, C2.2Dic24, C4.1Dic12, C22.8D24, (C2×C16).3S3, (C2×C48).5C2, (C2×C6).14D8, C3⋊2(C16⋊3C4), C24.69(C2×C4), (C2×C4).71D12, (C2×C8).297D6, C12.23(C4⋊C4), C24⋊1C4.2C2, C6.7(C2.D8), (C2×C12).370D4, C4.8(C4⋊Dic3), C8.14(C2×Dic3), C2.3(C24⋊1C4), (C2×C24).370C22, SmallGroup(192,63)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊5C4
G = < a,b | a48=b4=1, bab-1=a-1 >
Smallest permutation representation of C48⋊5C4
►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)
(1 106 71 145)(2 105 72 192)(3 104 73 191)(4 103 74 190)(5 102 75 189)(6 101 76 188)(7 100 77 187)(8 99 78 186)(9 98 79 185)(10 97 80 184)(11 144 81 183)(12 143 82 182)(13 142 83 181)(14 141 84 180)(15 140 85 179)(16 139 86 178)(17 138 87 177)(18 137 88 176)(19 136 89 175)(20 135 90 174)(21 134 91 173)(22 133 92 172)(23 132 93 171)(24 131 94 170)(25 130 95 169)(26 129 96 168)(27 128 49 167)(28 127 50 166)(29 126 51 165)(30 125 52 164)(31 124 53 163)(32 123 54 162)(33 122 55 161)(34 121 56 160)(35 120 57 159)(36 119 58 158)(37 118 59 157)(38 117 60 156)(39 116 61 155)(40 115 62 154)(41 114 63 153)(42 113 64 152)(43 112 65 151)(44 111 66 150)(45 110 67 149)(46 109 68 148)(47 108 69 147)(48 107 70 146)
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), (1,106,71,145)(2,105,72,192)(3,104,73,191)(4,103,74,190)(5,102,75,189)(6,101,76,188)(7,100,77,187)(8,99,78,186)(9,98,79,185)(10,97,80,184)(11,144,81,183)(12,143,82,182)(13,142,83,181)(14,141,84,180)(15,140,85,179)(16,139,86,178)(17,138,87,177)(18,137,88,176)(19,136,89,175)(20,135,90,174)(21,134,91,173)(22,133,92,172)(23,132,93,171)(24,131,94,170)(25,130,95,169)(26,129,96,168)(27,128,49,167)(28,127,50,166)(29,126,51,165)(30,125,52,164)(31,124,53,163)(32,123,54,162)(33,122,55,161)(34,121,56,160)(35,120,57,159)(36,119,58,158)(37,118,59,157)(38,117,60,156)(39,116,61,155)(40,115,62,154)(41,114,63,153)(42,113,64,152)(43,112,65,151)(44,111,66,150)(45,110,67,149)(46,109,68,148)(47,108,69,147)(48,107,70,146)>;
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), (1,106,71,145)(2,105,72,192)(3,104,73,191)(4,103,74,190)(5,102,75,189)(6,101,76,188)(7,100,77,187)(8,99,78,186)(9,98,79,185)(10,97,80,184)(11,144,81,183)(12,143,82,182)(13,142,83,181)(14,141,84,180)(15,140,85,179)(16,139,86,178)(17,138,87,177)(18,137,88,176)(19,136,89,175)(20,135,90,174)(21,134,91,173)(22,133,92,172)(23,132,93,171)(24,131,94,170)(25,130,95,169)(26,129,96,168)(27,128,49,167)(28,127,50,166)(29,126,51,165)(30,125,52,164)(31,124,53,163)(32,123,54,162)(33,122,55,161)(34,121,56,160)(35,120,57,159)(36,119,58,158)(37,118,59,157)(38,117,60,156)(39,116,61,155)(40,115,62,154)(41,114,63,153)(42,113,64,152)(43,112,65,151)(44,111,66,150)(45,110,67,149)(46,109,68,148)(47,108,69,147)(48,107,70,146) );
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)], [(1,106,71,145),(2,105,72,192),(3,104,73,191),(4,103,74,190),(5,102,75,189),(6,101,76,188),(7,100,77,187),(8,99,78,186),(9,98,79,185),(10,97,80,184),(11,144,81,183),(12,143,82,182),(13,142,83,181),(14,141,84,180),(15,140,85,179),(16,139,86,178),(17,138,87,177),(18,137,88,176),(19,136,89,175),(20,135,90,174),(21,134,91,173),(22,133,92,172),(23,132,93,171),(24,131,94,170),(25,130,95,169),(26,129,96,168),(27,128,49,167),(28,127,50,166),(29,126,51,165),(30,125,52,164),(31,124,53,163),(32,123,54,162),(33,122,55,161),(34,121,56,160),(35,120,57,159),(36,119,58,158),(37,118,59,157),(38,117,60,156),(39,116,61,155),(40,115,62,154),(41,114,63,153),(42,113,64,152),(43,112,65,151),(44,111,66,150),(45,110,67,149),(46,109,68,148),(47,108,69,147),(48,107,70,146)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 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 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | + | + | - | - | + | + | - | |
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | Dic3 | D6 | Q16 | D8 | Dic6 | D12 | D16 | Q32 | Dic12 | D24 | D48 | Dic24 |
| kernel | C48⋊5C4 | C24⋊1C4 | C2×C48 | C48 | C2×C16 | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C48⋊5C4 ►in GL3(𝔽97) generated by
| 1 | 0 | 0 |
| 0 | 78 | 13 |
| 0 | 84 | 65 |
| 75 | 0 | 0 |
| 0 | 57 | 61 |
| 0 | 4 | 40 |
G:=sub<GL(3,GF(97))| [1,0,0,0,78,84,0,13,65],[75,0,0,0,57,4,0,61,40] >;
C48⋊5C4 in GAP, Magma, Sage, TeX
C_{48}\rtimes_5C_4 % in TeX
G:=Group("C48:5C4"); // GroupNames label
G:=SmallGroup(192,63);
// by ID
G=gap.SmallGroup(192,63);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^-1>;
// generators/relations
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