metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C76.1C4, C4.Dic19, C4.15D38, C19⋊2M4(2), C22.Dic19, C76.15C22, C19⋊C8⋊5C2, (C2×C38).3C4, C38.7(C2×C4), (C2×C76).5C2, (C2×C4).2D19, C2.3(C2×Dic19), SmallGroup(304,9)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C76.C4
G = < a,b | a76=1, b4=a38, bab-1=a-1 >
Smallest permutation representation of C76.C4
►On 152 points
Generators in S152
(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)
(1 99 58 118 39 137 20 80)(2 98 59 117 40 136 21 79)(3 97 60 116 41 135 22 78)(4 96 61 115 42 134 23 77)(5 95 62 114 43 133 24 152)(6 94 63 113 44 132 25 151)(7 93 64 112 45 131 26 150)(8 92 65 111 46 130 27 149)(9 91 66 110 47 129 28 148)(10 90 67 109 48 128 29 147)(11 89 68 108 49 127 30 146)(12 88 69 107 50 126 31 145)(13 87 70 106 51 125 32 144)(14 86 71 105 52 124 33 143)(15 85 72 104 53 123 34 142)(16 84 73 103 54 122 35 141)(17 83 74 102 55 121 36 140)(18 82 75 101 56 120 37 139)(19 81 76 100 57 119 38 138)
G:=sub<Sym(152)| (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), (1,99,58,118,39,137,20,80)(2,98,59,117,40,136,21,79)(3,97,60,116,41,135,22,78)(4,96,61,115,42,134,23,77)(5,95,62,114,43,133,24,152)(6,94,63,113,44,132,25,151)(7,93,64,112,45,131,26,150)(8,92,65,111,46,130,27,149)(9,91,66,110,47,129,28,148)(10,90,67,109,48,128,29,147)(11,89,68,108,49,127,30,146)(12,88,69,107,50,126,31,145)(13,87,70,106,51,125,32,144)(14,86,71,105,52,124,33,143)(15,85,72,104,53,123,34,142)(16,84,73,103,54,122,35,141)(17,83,74,102,55,121,36,140)(18,82,75,101,56,120,37,139)(19,81,76,100,57,119,38,138)>;
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), (1,99,58,118,39,137,20,80)(2,98,59,117,40,136,21,79)(3,97,60,116,41,135,22,78)(4,96,61,115,42,134,23,77)(5,95,62,114,43,133,24,152)(6,94,63,113,44,132,25,151)(7,93,64,112,45,131,26,150)(8,92,65,111,46,130,27,149)(9,91,66,110,47,129,28,148)(10,90,67,109,48,128,29,147)(11,89,68,108,49,127,30,146)(12,88,69,107,50,126,31,145)(13,87,70,106,51,125,32,144)(14,86,71,105,52,124,33,143)(15,85,72,104,53,123,34,142)(16,84,73,103,54,122,35,141)(17,83,74,102,55,121,36,140)(18,82,75,101,56,120,37,139)(19,81,76,100,57,119,38,138) );
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)], [(1,99,58,118,39,137,20,80),(2,98,59,117,40,136,21,79),(3,97,60,116,41,135,22,78),(4,96,61,115,42,134,23,77),(5,95,62,114,43,133,24,152),(6,94,63,113,44,132,25,151),(7,93,64,112,45,131,26,150),(8,92,65,111,46,130,27,149),(9,91,66,110,47,129,28,148),(10,90,67,109,48,128,29,147),(11,89,68,108,49,127,30,146),(12,88,69,107,50,126,31,145),(13,87,70,106,51,125,32,144),(14,86,71,105,52,124,33,143),(15,85,72,104,53,123,34,142),(16,84,73,103,54,122,35,141),(17,83,74,102,55,121,36,140),(18,82,75,101,56,120,37,139),(19,81,76,100,57,119,38,138)]])
82 conjugacy classes
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 19A | ··· | 19I | 38A | ··· | 38AA | 76A | ··· | 76AJ |
| order | 1 | 2 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 8 | 19 | ··· | 19 | 38 | ··· | 38 | 76 | ··· | 76 |
| size | 1 | 1 | 2 | 1 | 1 | 2 | 38 | 38 | 38 | 38 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
82 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | ||||
| image | C1 | C2 | C2 | C4 | C4 | M4(2) | D19 | Dic19 | D38 | Dic19 | C76.C4 |
| kernel | C76.C4 | C19⋊C8 | C2×C76 | C76 | C2×C38 | C19 | C2×C4 | C4 | C4 | C22 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 9 | 9 | 9 | 9 | 36 |
Matrix representation of C76.C4 ►in GL2(𝔽457) generated by
| 220 | 0 |
| 29 | 430 |
| 139 | 455 |
| 9 | 318 |
G:=sub<GL(2,GF(457))| [220,29,0,430],[139,9,455,318] >;
C76.C4 in GAP, Magma, Sage, TeX
C_{76}.C_4 % in TeX
G:=Group("C76.C4"); // GroupNames label
G:=SmallGroup(304,9);
// by ID
G=gap.SmallGroup(304,9);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-19,20,101,42,7204]);
// Polycyclic
G:=Group<a,b|a^76=1,b^4=a^38,b*a*b^-1=a^-1>;
// generators/relations
Export