direct product, metacyclic, supersoluble, monomial, Z-group, 3-hyperelementary
Aliases: C7⋊C3×C23, C7⋊C69, C161⋊C3, SmallGroup(483,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C7 — C7⋊C3×C23 |
Generators and relations for C7⋊C3×C23
G = < a,b,c | a23=b7=c3=1, ab=ba, ac=ca, cbc-1=b4 >
Smallest permutation representation of C7⋊C3×C23
►On 161 points
Generators in S161
(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)
(1 94 91 147 47 137 26)(2 95 92 148 48 138 27)(3 96 70 149 49 116 28)(4 97 71 150 50 117 29)(5 98 72 151 51 118 30)(6 99 73 152 52 119 31)(7 100 74 153 53 120 32)(8 101 75 154 54 121 33)(9 102 76 155 55 122 34)(10 103 77 156 56 123 35)(11 104 78 157 57 124 36)(12 105 79 158 58 125 37)(13 106 80 159 59 126 38)(14 107 81 160 60 127 39)(15 108 82 161 61 128 40)(16 109 83 139 62 129 41)(17 110 84 140 63 130 42)(18 111 85 141 64 131 43)(19 112 86 142 65 132 44)(20 113 87 143 66 133 45)(21 114 88 144 67 134 46)(22 115 89 145 68 135 24)(23 93 90 146 69 136 25)
(24 135 145)(25 136 146)(26 137 147)(27 138 148)(28 116 149)(29 117 150)(30 118 151)(31 119 152)(32 120 153)(33 121 154)(34 122 155)(35 123 156)(36 124 157)(37 125 158)(38 126 159)(39 127 160)(40 128 161)(41 129 139)(42 130 140)(43 131 141)(44 132 142)(45 133 143)(46 134 144)(47 94 91)(48 95 92)(49 96 70)(50 97 71)(51 98 72)(52 99 73)(53 100 74)(54 101 75)(55 102 76)(56 103 77)(57 104 78)(58 105 79)(59 106 80)(60 107 81)(61 108 82)(62 109 83)(63 110 84)(64 111 85)(65 112 86)(66 113 87)(67 114 88)(68 115 89)(69 93 90)
G:=sub<Sym(161)| (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), (1,94,91,147,47,137,26)(2,95,92,148,48,138,27)(3,96,70,149,49,116,28)(4,97,71,150,50,117,29)(5,98,72,151,51,118,30)(6,99,73,152,52,119,31)(7,100,74,153,53,120,32)(8,101,75,154,54,121,33)(9,102,76,155,55,122,34)(10,103,77,156,56,123,35)(11,104,78,157,57,124,36)(12,105,79,158,58,125,37)(13,106,80,159,59,126,38)(14,107,81,160,60,127,39)(15,108,82,161,61,128,40)(16,109,83,139,62,129,41)(17,110,84,140,63,130,42)(18,111,85,141,64,131,43)(19,112,86,142,65,132,44)(20,113,87,143,66,133,45)(21,114,88,144,67,134,46)(22,115,89,145,68,135,24)(23,93,90,146,69,136,25), (24,135,145)(25,136,146)(26,137,147)(27,138,148)(28,116,149)(29,117,150)(30,118,151)(31,119,152)(32,120,153)(33,121,154)(34,122,155)(35,123,156)(36,124,157)(37,125,158)(38,126,159)(39,127,160)(40,128,161)(41,129,139)(42,130,140)(43,131,141)(44,132,142)(45,133,143)(46,134,144)(47,94,91)(48,95,92)(49,96,70)(50,97,71)(51,98,72)(52,99,73)(53,100,74)(54,101,75)(55,102,76)(56,103,77)(57,104,78)(58,105,79)(59,106,80)(60,107,81)(61,108,82)(62,109,83)(63,110,84)(64,111,85)(65,112,86)(66,113,87)(67,114,88)(68,115,89)(69,93,90)>;
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), (1,94,91,147,47,137,26)(2,95,92,148,48,138,27)(3,96,70,149,49,116,28)(4,97,71,150,50,117,29)(5,98,72,151,51,118,30)(6,99,73,152,52,119,31)(7,100,74,153,53,120,32)(8,101,75,154,54,121,33)(9,102,76,155,55,122,34)(10,103,77,156,56,123,35)(11,104,78,157,57,124,36)(12,105,79,158,58,125,37)(13,106,80,159,59,126,38)(14,107,81,160,60,127,39)(15,108,82,161,61,128,40)(16,109,83,139,62,129,41)(17,110,84,140,63,130,42)(18,111,85,141,64,131,43)(19,112,86,142,65,132,44)(20,113,87,143,66,133,45)(21,114,88,144,67,134,46)(22,115,89,145,68,135,24)(23,93,90,146,69,136,25), (24,135,145)(25,136,146)(26,137,147)(27,138,148)(28,116,149)(29,117,150)(30,118,151)(31,119,152)(32,120,153)(33,121,154)(34,122,155)(35,123,156)(36,124,157)(37,125,158)(38,126,159)(39,127,160)(40,128,161)(41,129,139)(42,130,140)(43,131,141)(44,132,142)(45,133,143)(46,134,144)(47,94,91)(48,95,92)(49,96,70)(50,97,71)(51,98,72)(52,99,73)(53,100,74)(54,101,75)(55,102,76)(56,103,77)(57,104,78)(58,105,79)(59,106,80)(60,107,81)(61,108,82)(62,109,83)(63,110,84)(64,111,85)(65,112,86)(66,113,87)(67,114,88)(68,115,89)(69,93,90) );
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)], [(1,94,91,147,47,137,26),(2,95,92,148,48,138,27),(3,96,70,149,49,116,28),(4,97,71,150,50,117,29),(5,98,72,151,51,118,30),(6,99,73,152,52,119,31),(7,100,74,153,53,120,32),(8,101,75,154,54,121,33),(9,102,76,155,55,122,34),(10,103,77,156,56,123,35),(11,104,78,157,57,124,36),(12,105,79,158,58,125,37),(13,106,80,159,59,126,38),(14,107,81,160,60,127,39),(15,108,82,161,61,128,40),(16,109,83,139,62,129,41),(17,110,84,140,63,130,42),(18,111,85,141,64,131,43),(19,112,86,142,65,132,44),(20,113,87,143,66,133,45),(21,114,88,144,67,134,46),(22,115,89,145,68,135,24),(23,93,90,146,69,136,25)], [(24,135,145),(25,136,146),(26,137,147),(27,138,148),(28,116,149),(29,117,150),(30,118,151),(31,119,152),(32,120,153),(33,121,154),(34,122,155),(35,123,156),(36,124,157),(37,125,158),(38,126,159),(39,127,160),(40,128,161),(41,129,139),(42,130,140),(43,131,141),(44,132,142),(45,133,143),(46,134,144),(47,94,91),(48,95,92),(49,96,70),(50,97,71),(51,98,72),(52,99,73),(53,100,74),(54,101,75),(55,102,76),(56,103,77),(57,104,78),(58,105,79),(59,106,80),(60,107,81),(61,108,82),(62,109,83),(63,110,84),(64,111,85),(65,112,86),(66,113,87),(67,114,88),(68,115,89),(69,93,90)]])
115 conjugacy classes
| class | 1 | 3A | 3B | 7A | 7B | 23A | ··· | 23V | 69A | ··· | 69AR | 161A | ··· | 161AR |
| order | 1 | 3 | 3 | 7 | 7 | 23 | ··· | 23 | 69 | ··· | 69 | 161 | ··· | 161 |
| size | 1 | 7 | 7 | 3 | 3 | 1 | ··· | 1 | 7 | ··· | 7 | 3 | ··· | 3 |
115 irreducible representations
| dim | 1 | 1 | 1 | 1 | 3 | 3 |
| type | + | |||||
| image | C1 | C3 | C23 | C69 | C7⋊C3 | C7⋊C3×C23 |
| kernel | C7⋊C3×C23 | C161 | C7⋊C3 | C7 | C23 | C1 |
| # reps | 1 | 2 | 22 | 44 | 2 | 44 |
Matrix representation of C7⋊C3×C23 ►in GL3(𝔽967) generated by
| 69 | 0 | 0 |
| 0 | 69 | 0 |
| 0 | 0 | 69 |
| 966 | 966 | 699 |
| 1 | 0 | 1 |
| 0 | 1 | 268 |
| 0 | 700 | 1 |
| 1 | 1 | 268 |
| 0 | 268 | 966 |
G:=sub<GL(3,GF(967))| [69,0,0,0,69,0,0,0,69],[966,1,0,966,0,1,699,1,268],[0,1,0,700,1,268,1,268,966] >;
C7⋊C3×C23 in GAP, Magma, Sage, TeX
C_7\rtimes C_3\times C_{23} % in TeX
G:=Group("C7:C3xC23"); // GroupNames label
G:=SmallGroup(483,1);
// by ID
G=gap.SmallGroup(483,1);
# by ID
G:=PCGroup([3,-3,-23,-7,1244]);
// Polycyclic
G:=Group<a,b,c|a^23=b^7=c^3=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
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