metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C48⋊6C4, C16⋊4Dic3, C24.12Q8, C6.2SD32, C12.13Q16, C4.2Dic12, C8.11Dic6, C22.9D24, (C2×C48).8C2, (C2×C16).6S3, (C2×C6).15D8, C3⋊2(C16⋊4C4), C24.70(C2×C4), (C2×C8).298D6, (C2×C4).72D12, C12.24(C4⋊C4), C24⋊1C4.3C2, C6.8(C2.D8), (C2×C12).371D4, C4.9(C4⋊Dic3), C8.15(C2×Dic3), C2.2(C48⋊C2), C2.4(C24⋊1C4), (C2×C24).371C22, SmallGroup(192,64)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C48⋊6C4
G = < a,b | a48=b4=1, bab-1=a23 >
Smallest permutation representation of C48⋊6C4
►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)
(1 144 169 65)(2 119 170 88)(3 142 171 63)(4 117 172 86)(5 140 173 61)(6 115 174 84)(7 138 175 59)(8 113 176 82)(9 136 177 57)(10 111 178 80)(11 134 179 55)(12 109 180 78)(13 132 181 53)(14 107 182 76)(15 130 183 51)(16 105 184 74)(17 128 185 49)(18 103 186 72)(19 126 187 95)(20 101 188 70)(21 124 189 93)(22 99 190 68)(23 122 191 91)(24 97 192 66)(25 120 145 89)(26 143 146 64)(27 118 147 87)(28 141 148 62)(29 116 149 85)(30 139 150 60)(31 114 151 83)(32 137 152 58)(33 112 153 81)(34 135 154 56)(35 110 155 79)(36 133 156 54)(37 108 157 77)(38 131 158 52)(39 106 159 75)(40 129 160 50)(41 104 161 73)(42 127 162 96)(43 102 163 71)(44 125 164 94)(45 100 165 69)(46 123 166 92)(47 98 167 67)(48 121 168 90)
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), (1,144,169,65)(2,119,170,88)(3,142,171,63)(4,117,172,86)(5,140,173,61)(6,115,174,84)(7,138,175,59)(8,113,176,82)(9,136,177,57)(10,111,178,80)(11,134,179,55)(12,109,180,78)(13,132,181,53)(14,107,182,76)(15,130,183,51)(16,105,184,74)(17,128,185,49)(18,103,186,72)(19,126,187,95)(20,101,188,70)(21,124,189,93)(22,99,190,68)(23,122,191,91)(24,97,192,66)(25,120,145,89)(26,143,146,64)(27,118,147,87)(28,141,148,62)(29,116,149,85)(30,139,150,60)(31,114,151,83)(32,137,152,58)(33,112,153,81)(34,135,154,56)(35,110,155,79)(36,133,156,54)(37,108,157,77)(38,131,158,52)(39,106,159,75)(40,129,160,50)(41,104,161,73)(42,127,162,96)(43,102,163,71)(44,125,164,94)(45,100,165,69)(46,123,166,92)(47,98,167,67)(48,121,168,90)>;
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), (1,144,169,65)(2,119,170,88)(3,142,171,63)(4,117,172,86)(5,140,173,61)(6,115,174,84)(7,138,175,59)(8,113,176,82)(9,136,177,57)(10,111,178,80)(11,134,179,55)(12,109,180,78)(13,132,181,53)(14,107,182,76)(15,130,183,51)(16,105,184,74)(17,128,185,49)(18,103,186,72)(19,126,187,95)(20,101,188,70)(21,124,189,93)(22,99,190,68)(23,122,191,91)(24,97,192,66)(25,120,145,89)(26,143,146,64)(27,118,147,87)(28,141,148,62)(29,116,149,85)(30,139,150,60)(31,114,151,83)(32,137,152,58)(33,112,153,81)(34,135,154,56)(35,110,155,79)(36,133,156,54)(37,108,157,77)(38,131,158,52)(39,106,159,75)(40,129,160,50)(41,104,161,73)(42,127,162,96)(43,102,163,71)(44,125,164,94)(45,100,165,69)(46,123,166,92)(47,98,167,67)(48,121,168,90) );
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)], [(1,144,169,65),(2,119,170,88),(3,142,171,63),(4,117,172,86),(5,140,173,61),(6,115,174,84),(7,138,175,59),(8,113,176,82),(9,136,177,57),(10,111,178,80),(11,134,179,55),(12,109,180,78),(13,132,181,53),(14,107,182,76),(15,130,183,51),(16,105,184,74),(17,128,185,49),(18,103,186,72),(19,126,187,95),(20,101,188,70),(21,124,189,93),(22,99,190,68),(23,122,191,91),(24,97,192,66),(25,120,145,89),(26,143,146,64),(27,118,147,87),(28,141,148,62),(29,116,149,85),(30,139,150,60),(31,114,151,83),(32,137,152,58),(33,112,153,81),(34,135,154,56),(35,110,155,79),(36,133,156,54),(37,108,157,77),(38,131,158,52),(39,106,159,75),(40,129,160,50),(41,104,161,73),(42,127,162,96),(43,102,163,71),(44,125,164,94),(45,100,165,69),(46,123,166,92),(47,98,167,67),(48,121,168,90)]])
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 |
| type | + | + | + | + | - | + | - | + | - | + | - | + | - | + | |||
| image | C1 | C2 | C2 | C4 | S3 | Q8 | D4 | Dic3 | D6 | Q16 | D8 | Dic6 | D12 | SD32 | Dic12 | D24 | C48⋊C2 |
| kernel | C48⋊6C4 | C24⋊1C4 | C2×C48 | C48 | C2×C16 | C24 | C2×C12 | C16 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of C48⋊6C4 ►in GL4(𝔽97) generated by
| 2 | 16 | 0 | 0 |
| 81 | 18 | 0 | 0 |
| 0 | 0 | 95 | 43 |
| 0 | 0 | 54 | 52 |
| 94 | 21 | 0 | 0 |
| 18 | 3 | 0 | 0 |
| 0 | 0 | 73 | 53 |
| 0 | 0 | 77 | 24 |
G:=sub<GL(4,GF(97))| [2,81,0,0,16,18,0,0,0,0,95,54,0,0,43,52],[94,18,0,0,21,3,0,0,0,0,73,77,0,0,53,24] >;
C48⋊6C4 in GAP, Magma, Sage, TeX
C_{48}\rtimes_6C_4 % in TeX
G:=Group("C48:6C4"); // GroupNames label
G:=SmallGroup(192,64);
// by ID
G=gap.SmallGroup(192,64);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,176,1571,80,1684,102,6278]);
// Polycyclic
G:=Group<a,b|a^48=b^4=1,b*a*b^-1=a^23>;
// generators/relations
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