direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: M4(2)×C23, C4.C92, C8⋊3C46, C184⋊7C2, C92.4C4, C22.C92, C92.22C22, (C2×C46).1C4, (C2×C92).8C2, C2.3(C2×C92), (C2×C4).2C46, C4.6(C2×C46), C46.12(C2×C4), SmallGroup(368,23)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)×C23
G = < a,b,c | a23=b8=c2=1, ab=ba, ac=ca, cbc=b5 >
Smallest permutation representation of M4(2)×C23
►On 184 points
Generators in S184
(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)
(1 121 64 27 103 148 167 82)(2 122 65 28 104 149 168 83)(3 123 66 29 105 150 169 84)(4 124 67 30 106 151 170 85)(5 125 68 31 107 152 171 86)(6 126 69 32 108 153 172 87)(7 127 47 33 109 154 173 88)(8 128 48 34 110 155 174 89)(9 129 49 35 111 156 175 90)(10 130 50 36 112 157 176 91)(11 131 51 37 113 158 177 92)(12 132 52 38 114 159 178 70)(13 133 53 39 115 160 179 71)(14 134 54 40 93 161 180 72)(15 135 55 41 94 139 181 73)(16 136 56 42 95 140 182 74)(17 137 57 43 96 141 183 75)(18 138 58 44 97 142 184 76)(19 116 59 45 98 143 162 77)(20 117 60 46 99 144 163 78)(21 118 61 24 100 145 164 79)(22 119 62 25 101 146 165 80)(23 120 63 26 102 147 166 81)
(24 79)(25 80)(26 81)(27 82)(28 83)(29 84)(30 85)(31 86)(32 87)(33 88)(34 89)(35 90)(36 91)(37 92)(38 70)(39 71)(40 72)(41 73)(42 74)(43 75)(44 76)(45 77)(46 78)(116 143)(117 144)(118 145)(119 146)(120 147)(121 148)(122 149)(123 150)(124 151)(125 152)(126 153)(127 154)(128 155)(129 156)(130 157)(131 158)(132 159)(133 160)(134 161)(135 139)(136 140)(137 141)(138 142)
G:=sub<Sym(184)| (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), (1,121,64,27,103,148,167,82)(2,122,65,28,104,149,168,83)(3,123,66,29,105,150,169,84)(4,124,67,30,106,151,170,85)(5,125,68,31,107,152,171,86)(6,126,69,32,108,153,172,87)(7,127,47,33,109,154,173,88)(8,128,48,34,110,155,174,89)(9,129,49,35,111,156,175,90)(10,130,50,36,112,157,176,91)(11,131,51,37,113,158,177,92)(12,132,52,38,114,159,178,70)(13,133,53,39,115,160,179,71)(14,134,54,40,93,161,180,72)(15,135,55,41,94,139,181,73)(16,136,56,42,95,140,182,74)(17,137,57,43,96,141,183,75)(18,138,58,44,97,142,184,76)(19,116,59,45,98,143,162,77)(20,117,60,46,99,144,163,78)(21,118,61,24,100,145,164,79)(22,119,62,25,101,146,165,80)(23,120,63,26,102,147,166,81), (24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,139)(136,140)(137,141)(138,142)>;
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), (1,121,64,27,103,148,167,82)(2,122,65,28,104,149,168,83)(3,123,66,29,105,150,169,84)(4,124,67,30,106,151,170,85)(5,125,68,31,107,152,171,86)(6,126,69,32,108,153,172,87)(7,127,47,33,109,154,173,88)(8,128,48,34,110,155,174,89)(9,129,49,35,111,156,175,90)(10,130,50,36,112,157,176,91)(11,131,51,37,113,158,177,92)(12,132,52,38,114,159,178,70)(13,133,53,39,115,160,179,71)(14,134,54,40,93,161,180,72)(15,135,55,41,94,139,181,73)(16,136,56,42,95,140,182,74)(17,137,57,43,96,141,183,75)(18,138,58,44,97,142,184,76)(19,116,59,45,98,143,162,77)(20,117,60,46,99,144,163,78)(21,118,61,24,100,145,164,79)(22,119,62,25,101,146,165,80)(23,120,63,26,102,147,166,81), (24,79)(25,80)(26,81)(27,82)(28,83)(29,84)(30,85)(31,86)(32,87)(33,88)(34,89)(35,90)(36,91)(37,92)(38,70)(39,71)(40,72)(41,73)(42,74)(43,75)(44,76)(45,77)(46,78)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,139)(136,140)(137,141)(138,142) );
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)], [(1,121,64,27,103,148,167,82),(2,122,65,28,104,149,168,83),(3,123,66,29,105,150,169,84),(4,124,67,30,106,151,170,85),(5,125,68,31,107,152,171,86),(6,126,69,32,108,153,172,87),(7,127,47,33,109,154,173,88),(8,128,48,34,110,155,174,89),(9,129,49,35,111,156,175,90),(10,130,50,36,112,157,176,91),(11,131,51,37,113,158,177,92),(12,132,52,38,114,159,178,70),(13,133,53,39,115,160,179,71),(14,134,54,40,93,161,180,72),(15,135,55,41,94,139,181,73),(16,136,56,42,95,140,182,74),(17,137,57,43,96,141,183,75),(18,138,58,44,97,142,184,76),(19,116,59,45,98,143,162,77),(20,117,60,46,99,144,163,78),(21,118,61,24,100,145,164,79),(22,119,62,25,101,146,165,80),(23,120,63,26,102,147,166,81)], [(24,79),(25,80),(26,81),(27,82),(28,83),(29,84),(30,85),(31,86),(32,87),(33,88),(34,89),(35,90),(36,91),(37,92),(38,70),(39,71),(40,72),(41,73),(42,74),(43,75),(44,76),(45,77),(46,78),(116,143),(117,144),(118,145),(119,146),(120,147),(121,148),(122,149),(123,150),(124,151),(125,152),(126,153),(127,154),(128,155),(129,156),(130,157),(131,158),(132,159),(133,160),(134,161),(135,139),(136,140),(137,141),(138,142)]])
230 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 23A | ··· | 23V | 46A | ··· | 46V | 46W | ··· | 46AR | 92A | ··· | 92AR | 92AS | ··· | 92BN | 184A | ··· | 184CJ |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 23 | ··· | 23 | 46 | ··· | 46 | 46 | ··· | 46 | 92 | ··· | 92 | 92 | ··· | 92 | 184 | ··· | 184 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
230 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | |||||||||
| image | C1 | C2 | C2 | C4 | C4 | C23 | C46 | C46 | C92 | C92 | M4(2) | M4(2)×C23 |
| kernel | M4(2)×C23 | C184 | C2×C92 | C92 | C2×C46 | M4(2) | C8 | C2×C4 | C4 | C22 | C23 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 22 | 44 | 22 | 44 | 44 | 2 | 44 |
Matrix representation of M4(2)×C23 ►in GL2(𝔽1289) generated by
| 669 | 0 |
| 0 | 669 |
| 5 | 457 |
| 331 | 1284 |
| 1 | 183 |
| 0 | 1288 |
G:=sub<GL(2,GF(1289))| [669,0,0,669],[5,331,457,1284],[1,0,183,1288] >;
M4(2)×C23 in GAP, Magma, Sage, TeX
M_4(2)\times C_{23} % in TeX
G:=Group("M4(2)xC23"); // GroupNames label
G:=SmallGroup(368,23);
// by ID
G=gap.SmallGroup(368,23);
# by ID
G:=PCGroup([5,-2,-2,-23,-2,-2,460,1861,58]);
// Polycyclic
G:=Group<a,b,c|a^23=b^8=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^5>;
// generators/relations
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