Copied to
clipboard

## G = C72.C6order 432 = 24·33

### 1st non-split extension by C72 of C6 acting faithfully

Series: Derived Chief Lower central Upper central

 Derived series C1 — C36 — C72.C6
 Chief series C1 — C3 — C9 — C18 — C36 — C4×3- 1+2 — C36.C6 — C72.C6
 Lower central C9 — C18 — C36 — C72.C6
 Upper central C1 — C2 — C4 — C8

Generators and relations for C72.C6
G = < a,b | a72=1, b6=a36, bab-1=a47 >

Subgroups: 222 in 58 conjugacy classes, 26 normal (22 characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C24, C24, Dic6, C3×Q8, 3- 1+2, Dic9, C36, C36, C3×Dic3, C3×C12, Dic12, C3×Q16, C2×3- 1+2, C72, C72, Dic18, C3×C24, C3×Dic6, C9⋊C12, C4×3- 1+2, Dic36, C3×Dic12, C8×3- 1+2, C36.C6, C72.C6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, Q16, C3×S3, D12, C3×D4, S3×C6, Dic12, C3×Q16, C9⋊C6, C3×D12, C2×C9⋊C6, C3×Dic12, D36⋊C3, C72.C6

Smallest permutation representation of C72.C6
On 144 points
Generators in S144
```(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)
(1 87 37 123)(2 110 62 122 50 134 38 74 26 86 14 98)(3 133 15 121 27 109 39 97 51 85 63 73)(4 84 40 120)(5 107 65 119 53 131 41 143 29 83 17 95)(6 130 18 118 30 106 42 94 54 82 66 142)(7 81 43 117)(8 104 68 116 56 128 44 140 32 80 20 92)(9 127 21 115 33 103 45 91 57 79 69 139)(10 78 46 114)(11 101 71 113 59 125 47 137 35 77 23 89)(12 124 24 112 36 100 48 88 60 76 72 136)(13 75 49 111)(16 144 52 108)(19 141 55 105)(22 138 58 102)(25 135 61 99)(28 132 64 96)(31 129 67 93)(34 126 70 90)```

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

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

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

53 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 9A 9B 9C 12A 12B 12C 12D 12E 12F 12G 12H 18A 18B 18C 24A 24B 24C 24D 24E 24F 24G 24H 36A ··· 36F 72A ··· 72L order 1 2 3 3 3 4 4 4 6 6 6 8 8 9 9 9 12 12 12 12 12 12 12 12 18 18 18 24 24 24 24 24 24 24 24 36 ··· 36 72 ··· 72 size 1 1 2 3 3 2 36 36 2 3 3 2 2 6 6 6 2 2 6 6 36 36 36 36 6 6 6 2 2 2 2 6 6 6 6 6 ··· 6 6 ··· 6

53 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 6 6 6 6 type + + + + + + - + - + + + - image C1 C2 C2 C3 C6 C6 S3 D4 D6 Q16 C3×S3 C3×D4 D12 S3×C6 C3×Q16 Dic12 C3×D12 C3×Dic12 C9⋊C6 C2×C9⋊C6 D36⋊C3 C72.C6 kernel C72.C6 C8×3- 1+2 C36.C6 Dic36 C72 Dic18 C3×C24 C2×3- 1+2 C3×C12 3- 1+2 C24 C18 C3×C6 C12 C9 C32 C6 C3 C8 C4 C2 C1 # reps 1 1 2 2 2 4 1 1 1 2 2 2 2 2 4 4 4 8 1 1 2 4

Matrix representation of C72.C6 in GL6(𝔽73)

 5 5 5 5 28 60 23 23 23 23 41 28 68 23 0 0 0 0 50 18 0 0 0 0 0 50 0 50 50 68 0 50 68 0 50 68
,
 68 19 0 0 0 0 14 5 0 0 0 0 54 54 54 54 40 49 68 68 68 68 49 9 5 0 0 19 19 5 5 0 19 5 19 5

`G:=sub<GL(6,GF(73))| [5,23,68,50,0,0,5,23,23,18,50,50,5,23,0,0,0,68,5,23,0,0,50,0,28,41,0,0,50,50,60,28,0,0,68,68],[68,14,54,68,5,5,19,5,54,68,0,0,0,0,54,68,0,19,0,0,54,68,19,5,0,0,40,49,19,19,0,0,49,9,5,5] >;`

C72.C6 in GAP, Magma, Sage, TeX

`C_{72}.C_6`
`% in TeX`

`G:=Group("C72.C6");`
`// GroupNames label`

`G:=SmallGroup(432,119);`
`// by ID`

`G=gap.SmallGroup(432,119);`
`# by ID`

`G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,168,197,260,1011,80,10085,2035,292,14118]);`
`// Polycyclic`

`G:=Group<a,b|a^72=1,b^6=a^36,b*a*b^-1=a^47>;`
`// generators/relations`

׿
×
𝔽