metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊10C4, C16⋊5Dic3, C6.2M5(2), C12.17C42, C3⋊C16⋊8C4, C3⋊C8.3C8, C6.8(C4×C8), C8.41(C4×S3), C4.21(S3×C8), (C2×C16).8S3, C3⋊2(C16⋊5C4), (C2×C48).16C2, C24.62(C2×C4), C12.26(C2×C8), (C2×C8).334D6, C2.4(C8×Dic3), C22.10(S3×C8), C2.2(D6.C8), (C8×Dic3).9C2, (C2×Dic3).3C8, C8.24(C2×Dic3), C4.16(C4×Dic3), (C4×Dic3).11C4, (C2×C24).419C22, (C2×C3⋊C8).13C4, (C2×C3⋊C16).10C2, (C2×C6).11(C2×C8), (C2×C4).168(C4×S3), (C2×C12).242(C2×C4), SmallGroup(192,61)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊10C4
G = < a,b | a48=b4=1, bab-1=a41 >
Smallest permutation representation of C48⋊10C4
►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 83 151)(2 99 84 192)(3 140 85 185)(4 133 86 178)(5 126 87 171)(6 119 88 164)(7 112 89 157)(8 105 90 150)(9 98 91 191)(10 139 92 184)(11 132 93 177)(12 125 94 170)(13 118 95 163)(14 111 96 156)(15 104 49 149)(16 97 50 190)(17 138 51 183)(18 131 52 176)(19 124 53 169)(20 117 54 162)(21 110 55 155)(22 103 56 148)(23 144 57 189)(24 137 58 182)(25 130 59 175)(26 123 60 168)(27 116 61 161)(28 109 62 154)(29 102 63 147)(30 143 64 188)(31 136 65 181)(32 129 66 174)(33 122 67 167)(34 115 68 160)(35 108 69 153)(36 101 70 146)(37 142 71 187)(38 135 72 180)(39 128 73 173)(40 121 74 166)(41 114 75 159)(42 107 76 152)(43 100 77 145)(44 141 78 186)(45 134 79 179)(46 127 80 172)(47 120 81 165)(48 113 82 158)
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), (1,106,83,151)(2,99,84,192)(3,140,85,185)(4,133,86,178)(5,126,87,171)(6,119,88,164)(7,112,89,157)(8,105,90,150)(9,98,91,191)(10,139,92,184)(11,132,93,177)(12,125,94,170)(13,118,95,163)(14,111,96,156)(15,104,49,149)(16,97,50,190)(17,138,51,183)(18,131,52,176)(19,124,53,169)(20,117,54,162)(21,110,55,155)(22,103,56,148)(23,144,57,189)(24,137,58,182)(25,130,59,175)(26,123,60,168)(27,116,61,161)(28,109,62,154)(29,102,63,147)(30,143,64,188)(31,136,65,181)(32,129,66,174)(33,122,67,167)(34,115,68,160)(35,108,69,153)(36,101,70,146)(37,142,71,187)(38,135,72,180)(39,128,73,173)(40,121,74,166)(41,114,75,159)(42,107,76,152)(43,100,77,145)(44,141,78,186)(45,134,79,179)(46,127,80,172)(47,120,81,165)(48,113,82,158)>;
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), (1,106,83,151)(2,99,84,192)(3,140,85,185)(4,133,86,178)(5,126,87,171)(6,119,88,164)(7,112,89,157)(8,105,90,150)(9,98,91,191)(10,139,92,184)(11,132,93,177)(12,125,94,170)(13,118,95,163)(14,111,96,156)(15,104,49,149)(16,97,50,190)(17,138,51,183)(18,131,52,176)(19,124,53,169)(20,117,54,162)(21,110,55,155)(22,103,56,148)(23,144,57,189)(24,137,58,182)(25,130,59,175)(26,123,60,168)(27,116,61,161)(28,109,62,154)(29,102,63,147)(30,143,64,188)(31,136,65,181)(32,129,66,174)(33,122,67,167)(34,115,68,160)(35,108,69,153)(36,101,70,146)(37,142,71,187)(38,135,72,180)(39,128,73,173)(40,121,74,166)(41,114,75,159)(42,107,76,152)(43,100,77,145)(44,141,78,186)(45,134,79,179)(46,127,80,172)(47,120,81,165)(48,113,82,158) );
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)], [(1,106,83,151),(2,99,84,192),(3,140,85,185),(4,133,86,178),(5,126,87,171),(6,119,88,164),(7,112,89,157),(8,105,90,150),(9,98,91,191),(10,139,92,184),(11,132,93,177),(12,125,94,170),(13,118,95,163),(14,111,96,156),(15,104,49,149),(16,97,50,190),(17,138,51,183),(18,131,52,176),(19,124,53,169),(20,117,54,162),(21,110,55,155),(22,103,56,148),(23,144,57,189),(24,137,58,182),(25,130,59,175),(26,123,60,168),(27,116,61,161),(28,109,62,154),(29,102,63,147),(30,143,64,188),(31,136,65,181),(32,129,66,174),(33,122,67,167),(34,115,68,160),(35,108,69,153),(36,101,70,146),(37,142,71,187),(38,135,72,180),(39,128,73,173),(40,121,74,166),(41,114,75,159),(42,107,76,152),(43,100,77,145),(44,141,78,186),(45,134,79,179),(46,127,80,172),(47,120,81,165),(48,113,82,158)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 16A | ··· | 16H | 16I | ··· | 16P | 24A | ··· | 24H | 48A | ··· | 48P |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 1 | ··· | 1 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | ··· | 2 | 2 | ··· | 2 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | C8 | C8 | S3 | Dic3 | D6 | C4×S3 | C4×S3 | M5(2) | S3×C8 | S3×C8 | D6.C8 |
| kernel | C48⋊10C4 | C2×C3⋊C16 | C8×Dic3 | C2×C48 | C3⋊C16 | C48 | C2×C3⋊C8 | C4×Dic3 | C3⋊C8 | C2×Dic3 | C2×C16 | C16 | C2×C8 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 8 | 8 | 1 | 2 | 1 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C48⋊10C4 ►in GL3(𝔽97) generated by
| 22 | 0 | 0 |
| 0 | 80 | 40 |
| 0 | 57 | 40 |
| 75 | 0 | 0 |
| 0 | 8 | 44 |
| 0 | 36 | 89 |
G:=sub<GL(3,GF(97))| [22,0,0,0,80,57,0,40,40],[75,0,0,0,8,36,0,44,89] >;
C48⋊10C4 in GAP, Magma, Sage, TeX
C_{48}\rtimes_{10}C_4 % in TeX
G:=Group("C48:10C4"); // GroupNames label
G:=SmallGroup(192,61);
// by ID
G=gap.SmallGroup(192,61);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,477,64,80,102,6278]);
// Polycyclic
G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^41>;
// generators/relations
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