direct product, abelian, monomial, 2-elementary
Aliases: C4×C48, SmallGroup(192,151)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C4×C48 |
| C1 — C4×C48 |
| C1 — C4×C48 |
Generators and relations for C4×C48
G = < a,b | a4=b48=1, ab=ba >
Smallest permutation representation of C4×C48
►Regular action on 192 points
Generators in S192
(1 157 122 52)(2 158 123 53)(3 159 124 54)(4 160 125 55)(5 161 126 56)(6 162 127 57)(7 163 128 58)(8 164 129 59)(9 165 130 60)(10 166 131 61)(11 167 132 62)(12 168 133 63)(13 169 134 64)(14 170 135 65)(15 171 136 66)(16 172 137 67)(17 173 138 68)(18 174 139 69)(19 175 140 70)(20 176 141 71)(21 177 142 72)(22 178 143 73)(23 179 144 74)(24 180 97 75)(25 181 98 76)(26 182 99 77)(27 183 100 78)(28 184 101 79)(29 185 102 80)(30 186 103 81)(31 187 104 82)(32 188 105 83)(33 189 106 84)(34 190 107 85)(35 191 108 86)(36 192 109 87)(37 145 110 88)(38 146 111 89)(39 147 112 90)(40 148 113 91)(41 149 114 92)(42 150 115 93)(43 151 116 94)(44 152 117 95)(45 153 118 96)(46 154 119 49)(47 155 120 50)(48 156 121 51)
(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)
G:=sub<Sym(192)| (1,157,122,52)(2,158,123,53)(3,159,124,54)(4,160,125,55)(5,161,126,56)(6,162,127,57)(7,163,128,58)(8,164,129,59)(9,165,130,60)(10,166,131,61)(11,167,132,62)(12,168,133,63)(13,169,134,64)(14,170,135,65)(15,171,136,66)(16,172,137,67)(17,173,138,68)(18,174,139,69)(19,175,140,70)(20,176,141,71)(21,177,142,72)(22,178,143,73)(23,179,144,74)(24,180,97,75)(25,181,98,76)(26,182,99,77)(27,183,100,78)(28,184,101,79)(29,185,102,80)(30,186,103,81)(31,187,104,82)(32,188,105,83)(33,189,106,84)(34,190,107,85)(35,191,108,86)(36,192,109,87)(37,145,110,88)(38,146,111,89)(39,147,112,90)(40,148,113,91)(41,149,114,92)(42,150,115,93)(43,151,116,94)(44,152,117,95)(45,153,118,96)(46,154,119,49)(47,155,120,50)(48,156,121,51), (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)>;
G:=Group( (1,157,122,52)(2,158,123,53)(3,159,124,54)(4,160,125,55)(5,161,126,56)(6,162,127,57)(7,163,128,58)(8,164,129,59)(9,165,130,60)(10,166,131,61)(11,167,132,62)(12,168,133,63)(13,169,134,64)(14,170,135,65)(15,171,136,66)(16,172,137,67)(17,173,138,68)(18,174,139,69)(19,175,140,70)(20,176,141,71)(21,177,142,72)(22,178,143,73)(23,179,144,74)(24,180,97,75)(25,181,98,76)(26,182,99,77)(27,183,100,78)(28,184,101,79)(29,185,102,80)(30,186,103,81)(31,187,104,82)(32,188,105,83)(33,189,106,84)(34,190,107,85)(35,191,108,86)(36,192,109,87)(37,145,110,88)(38,146,111,89)(39,147,112,90)(40,148,113,91)(41,149,114,92)(42,150,115,93)(43,151,116,94)(44,152,117,95)(45,153,118,96)(46,154,119,49)(47,155,120,50)(48,156,121,51), (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) );
G=PermutationGroup([[(1,157,122,52),(2,158,123,53),(3,159,124,54),(4,160,125,55),(5,161,126,56),(6,162,127,57),(7,163,128,58),(8,164,129,59),(9,165,130,60),(10,166,131,61),(11,167,132,62),(12,168,133,63),(13,169,134,64),(14,170,135,65),(15,171,136,66),(16,172,137,67),(17,173,138,68),(18,174,139,69),(19,175,140,70),(20,176,141,71),(21,177,142,72),(22,178,143,73),(23,179,144,74),(24,180,97,75),(25,181,98,76),(26,182,99,77),(27,183,100,78),(28,184,101,79),(29,185,102,80),(30,186,103,81),(31,187,104,82),(32,188,105,83),(33,189,106,84),(34,190,107,85),(35,191,108,86),(36,192,109,87),(37,145,110,88),(38,146,111,89),(39,147,112,90),(40,148,113,91),(41,149,114,92),(42,150,115,93),(43,151,116,94),(44,152,117,95),(45,153,118,96),(46,154,119,49),(47,155,120,50),(48,156,121,51)], [(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)]])
192 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4L | 6A | ··· | 6F | 8A | ··· | 8P | 12A | ··· | 12X | 16A | ··· | 16AF | 24A | ··· | 24AF | 48A | ··· | 48BL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 16 | ··· | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
192 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
| type | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C8 | C8 | C12 | C12 | C12 | C16 | C24 | C24 | C48 |
| kernel | C4×C48 | C4×C24 | C2×C48 | C4×C16 | C48 | C4×C12 | C2×C24 | C4×C8 | C2×C16 | C24 | C2×C12 | C16 | C42 | C2×C8 | C12 | C8 | C2×C4 | C4 |
| # reps | 1 | 1 | 2 | 2 | 8 | 2 | 2 | 2 | 4 | 8 | 8 | 16 | 4 | 4 | 32 | 16 | 16 | 64 |
Matrix representation of C4×C48 ►in GL3(𝔽97) generated by
| 22 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 22 |
| 70 | 0 | 0 |
| 0 | 24 | 0 |
| 0 | 0 | 33 |
G:=sub<GL(3,GF(97))| [22,0,0,0,1,0,0,0,22],[70,0,0,0,24,0,0,0,33] >;
C4×C48 in GAP, Magma, Sage, TeX
C_4\times C_{48} % in TeX
G:=Group("C4xC48"); // GroupNames label
G:=SmallGroup(192,151);
// by ID
G=gap.SmallGroup(192,151);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,84,176,136,124]);
// Polycyclic
G:=Group<a,b|a^4=b^48=1,a*b=b*a>;
// generators/relations
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