Copied to
clipboard

## G = C36.D6order 432 = 24·33

### 12nd non-split extension by C36 of D6 acting via D6/C3=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — C36.D6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×D12 — C36.D6
 Lower central C3×C9 — C3×C18 — C3×C36 — C36.D6
 Upper central C1 — C2 — C4

Generators and relations for C36.D6
G = < a,b,c | a36=1, b6=a27, c2=a18, bab-1=a17, cac-1=a-1, cbc-1=a27b5 >

Subgroups: 420 in 72 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×C9, Dic9, C36, C36, C2×C18, C3⋊Dic3, C3×C12, S3×C6, C24⋊C2, D4.S3, S3×C9, C3×C18, C9⋊C8, Dic18, D4×C9, C3×C3⋊C8, C3×D12, C324Q8, C9⋊Dic3, C3×C36, S3×C18, D4.D9, D12.S3, C3×C9⋊C8, C9×D12, C12.D9, C36.D6
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, D4.S3, C9⋊D4, C3⋊D12, S3×D9, D4.D9, D12.S3, C9⋊D12, C36.D6

Smallest permutation representation of C36.D6
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 49 22 46 7 43 28 40 13 37 34 70 19 67 4 64 25 61 10 58 31 55 16 52)(2 66 23 63 8 60 29 57 14 54 35 51 20 48 5 45 26 42 11 39 32 72 17 69)(3 47 24 44 9 41 30 38 15 71 36 68 21 65 6 62 27 59 12 56 33 53 18 50)(73 115 106 136 103 121 100 142 97 127 94 112 91 133 88 118 85 139 82 124 79 109 76 130)(74 132 107 117 104 138 101 123 98 144 95 129 92 114 89 135 86 120 83 141 80 126 77 111)(75 113 108 134 105 119 102 140 99 125 96 110 93 131 90 116 87 137 84 122 81 143 78 128)
(1 130 19 112)(2 129 20 111)(3 128 21 110)(4 127 22 109)(5 126 23 144)(6 125 24 143)(7 124 25 142)(8 123 26 141)(9 122 27 140)(10 121 28 139)(11 120 29 138)(12 119 30 137)(13 118 31 136)(14 117 32 135)(15 116 33 134)(16 115 34 133)(17 114 35 132)(18 113 36 131)(37 106 55 88)(38 105 56 87)(39 104 57 86)(40 103 58 85)(41 102 59 84)(42 101 60 83)(43 100 61 82)(44 99 62 81)(45 98 63 80)(46 97 64 79)(47 96 65 78)(48 95 66 77)(49 94 67 76)(50 93 68 75)(51 92 69 74)(52 91 70 73)(53 90 71 108)(54 89 72 107)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 18A 18B 18C 18D 18E 18F 18G ··· 18L 24A 24B 24C 24D 36A ··· 36I order 1 2 2 3 3 3 4 4 6 6 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 24 24 24 24 36 ··· 36 size 1 1 12 2 2 4 2 108 2 2 4 12 12 18 18 2 2 2 4 4 4 2 2 4 4 4 2 2 2 4 4 4 12 ··· 12 18 18 18 18 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + - + + - - + - image C1 C2 C2 C2 S3 S3 D4 D6 D6 SD16 D9 D12 C3⋊D4 D18 C24⋊C2 C9⋊D4 S32 D4.S3 C3⋊D12 S3×D9 D4.D9 D12.S3 C9⋊D12 C36.D6 kernel C36.D6 C3×C9⋊C8 C9×D12 C12.D9 C9⋊C8 C3×D12 C3×C18 C36 C3×C12 C3×C9 D12 C18 C3×C6 C12 C9 C6 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 3 2 2 3 4 6 1 1 1 3 3 2 3 6

Matrix representation of C36.D6 in GL6(𝔽73)

 7 30 0 0 0 0 47 66 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 70 59 0 0 0 0 56 42
,
 36 34 0 0 0 0 63 25 0 0 0 0 0 0 1 72 0 0 0 0 1 0 0 0 0 0 0 0 72 0 0 0 0 0 71 1
,
 12 68 0 0 0 0 29 61 0 0 0 0 0 0 1 0 0 0 0 0 1 72 0 0 0 0 0 0 1 0 0 0 0 0 2 72

`G:=sub<GL(6,GF(73))| [7,47,0,0,0,0,30,66,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,70,56,0,0,0,0,59,42],[36,63,0,0,0,0,34,25,0,0,0,0,0,0,1,1,0,0,0,0,72,0,0,0,0,0,0,0,72,71,0,0,0,0,0,1],[12,29,0,0,0,0,68,61,0,0,0,0,0,0,1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,2,0,0,0,0,0,72] >;`

C36.D6 in GAP, Magma, Sage, TeX

`C_{36}.D_6`
`% in TeX`

`G:=Group("C36.D6");`
`// GroupNames label`

`G:=SmallGroup(432,71);`
`// by ID`

`G=gap.SmallGroup(432,71);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=1,b^6=a^27,c^2=a^18,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^27*b^5>;`
`// generators/relations`

׿
×
𝔽