Copied to
clipboard

## G = C76.C4order 304 = 24·19

### 1st non-split extension by C76 of C4 acting via C4/C2=C2

Aliases: C76.1C4, C4.Dic19, C4.15D38, C192M4(2), C22.Dic19, C76.15C22, C19⋊C85C2, (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

 Derived series C1 — C38 — C76.C4
 Chief series C1 — C19 — C38 — C76 — C19⋊C8 — C76.C4
 Lower central C19 — C38 — C76.C4
 Upper central C1 — C4 — C2×C4

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 92 58 111 39 130 20 149)(2 91 59 110 40 129 21 148)(3 90 60 109 41 128 22 147)(4 89 61 108 42 127 23 146)(5 88 62 107 43 126 24 145)(6 87 63 106 44 125 25 144)(7 86 64 105 45 124 26 143)(8 85 65 104 46 123 27 142)(9 84 66 103 47 122 28 141)(10 83 67 102 48 121 29 140)(11 82 68 101 49 120 30 139)(12 81 69 100 50 119 31 138)(13 80 70 99 51 118 32 137)(14 79 71 98 52 117 33 136)(15 78 72 97 53 116 34 135)(16 77 73 96 54 115 35 134)(17 152 74 95 55 114 36 133)(18 151 75 94 56 113 37 132)(19 150 76 93 57 112 38 131)```

`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), (1,92,58,111,39,130,20,149)(2,91,59,110,40,129,21,148)(3,90,60,109,41,128,22,147)(4,89,61,108,42,127,23,146)(5,88,62,107,43,126,24,145)(6,87,63,106,44,125,25,144)(7,86,64,105,45,124,26,143)(8,85,65,104,46,123,27,142)(9,84,66,103,47,122,28,141)(10,83,67,102,48,121,29,140)(11,82,68,101,49,120,30,139)(12,81,69,100,50,119,31,138)(13,80,70,99,51,118,32,137)(14,79,71,98,52,117,33,136)(15,78,72,97,53,116,34,135)(16,77,73,96,54,115,35,134)(17,152,74,95,55,114,36,133)(18,151,75,94,56,113,37,132)(19,150,76,93,57,112,38,131)>;`

`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), (1,92,58,111,39,130,20,149)(2,91,59,110,40,129,21,148)(3,90,60,109,41,128,22,147)(4,89,61,108,42,127,23,146)(5,88,62,107,43,126,24,145)(6,87,63,106,44,125,25,144)(7,86,64,105,45,124,26,143)(8,85,65,104,46,123,27,142)(9,84,66,103,47,122,28,141)(10,83,67,102,48,121,29,140)(11,82,68,101,49,120,30,139)(12,81,69,100,50,119,31,138)(13,80,70,99,51,118,32,137)(14,79,71,98,52,117,33,136)(15,78,72,97,53,116,34,135)(16,77,73,96,54,115,35,134)(17,152,74,95,55,114,36,133)(18,151,75,94,56,113,37,132)(19,150,76,93,57,112,38,131) );`

`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)], [(1,92,58,111,39,130,20,149),(2,91,59,110,40,129,21,148),(3,90,60,109,41,128,22,147),(4,89,61,108,42,127,23,146),(5,88,62,107,43,126,24,145),(6,87,63,106,44,125,25,144),(7,86,64,105,45,124,26,143),(8,85,65,104,46,123,27,142),(9,84,66,103,47,122,28,141),(10,83,67,102,48,121,29,140),(11,82,68,101,49,120,30,139),(12,81,69,100,50,119,31,138),(13,80,70,99,51,118,32,137),(14,79,71,98,52,117,33,136),(15,78,72,97,53,116,34,135),(16,77,73,96,54,115,35,134),(17,152,74,95,55,114,36,133),(18,151,75,94,56,113,37,132),(19,150,76,93,57,112,38,131)])`

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

׿
×
𝔽