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## G = C36.9D4order 288 = 25·32

### 9th non-split extension by C36 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.9D4
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C4.Dic9 — C36.9D4
 Lower central C9 — C18 — C2×C18 — C36.9D4
 Upper central C1 — C2 — C2×C4 — C2×Q8

Generators and relations for C36.9D4
G = < a,b,c | a36=1, b4=a18, c2=a9, bab-1=a-1, cac-1=a17, cbc-1=a9b3 >

Subgroups: 148 in 57 conjugacy classes, 30 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, Q8, C9, C12, C12, C2×C6, M4(2), C2×Q8, C18, C18, C3⋊C8, C2×C12, C2×C12, C3×Q8, C4.10D4, C36, C36, C2×C18, C4.Dic3, C6×Q8, C9⋊C8, C2×C36, C2×C36, Q8×C9, C12.10D4, C4.Dic9, Q8×C18, C36.9D4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D9, C2×Dic3, C3⋊D4, C4.10D4, Dic9, D18, C6.D4, C2×Dic9, C9⋊D4, C12.10D4, C18.D4, C36.9D4

Smallest permutation representation of C36.9D4
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)
(1 135 28 144 19 117 10 126)(2 134 29 143 20 116 11 125)(3 133 30 142 21 115 12 124)(4 132 31 141 22 114 13 123)(5 131 32 140 23 113 14 122)(6 130 33 139 24 112 15 121)(7 129 34 138 25 111 16 120)(8 128 35 137 26 110 17 119)(9 127 36 136 27 109 18 118)(37 101 64 74 55 83 46 92)(38 100 65 73 56 82 47 91)(39 99 66 108 57 81 48 90)(40 98 67 107 58 80 49 89)(41 97 68 106 59 79 50 88)(42 96 69 105 60 78 51 87)(43 95 70 104 61 77 52 86)(44 94 71 103 62 76 53 85)(45 93 72 102 63 75 54 84)
(1 83 10 92 19 101 28 74)(2 100 11 73 20 82 29 91)(3 81 12 90 21 99 30 108)(4 98 13 107 22 80 31 89)(5 79 14 88 23 97 32 106)(6 96 15 105 24 78 33 87)(7 77 16 86 25 95 34 104)(8 94 17 103 26 76 35 85)(9 75 18 84 27 93 36 102)(37 135 46 144 55 117 64 126)(38 116 47 125 56 134 65 143)(39 133 48 142 57 115 66 124)(40 114 49 123 58 132 67 141)(41 131 50 140 59 113 68 122)(42 112 51 121 60 130 69 139)(43 129 52 138 61 111 70 120)(44 110 53 119 62 128 71 137)(45 127 54 136 63 109 72 118)```

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

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

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

51 conjugacy classes

 class 1 2A 2B 3 4A 4B 4C 4D 6A 6B 6C 8A 8B 8C 8D 9A 9B 9C 12A ··· 12F 18A ··· 18I 36A ··· 36R order 1 2 2 3 4 4 4 4 6 6 6 8 8 8 8 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 2 2 2 2 4 4 2 2 2 36 36 36 36 2 2 2 4 ··· 4 2 ··· 2 4 ··· 4

51 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + - + + - + - image C1 C2 C2 C4 S3 D4 Dic3 D6 D9 C3⋊D4 Dic9 D18 C9⋊D4 C4.10D4 C12.10D4 C36.9D4 kernel C36.9D4 C4.Dic9 Q8×C18 C2×C36 C6×Q8 C36 C2×C12 C2×C12 C2×Q8 C12 C2×C4 C2×C4 C4 C9 C3 C1 # reps 1 2 1 4 1 2 2 1 3 4 6 3 12 1 2 6

Matrix representation of C36.9D4 in GL6(𝔽73)

 41 65 0 0 0 0 0 57 0 0 0 0 0 0 0 8 0 0 0 0 65 0 0 0 0 0 0 0 0 64 0 0 0 0 9 0
,
 51 67 0 0 0 0 44 22 0 0 0 0 0 0 0 0 8 9 0 0 0 0 9 65 0 0 64 8 0 0 0 0 8 9 0 0
,
 51 57 0 0 0 0 44 22 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 72 0 0 0

`G:=sub<GL(6,GF(73))| [41,0,0,0,0,0,65,57,0,0,0,0,0,0,0,65,0,0,0,0,8,0,0,0,0,0,0,0,0,9,0,0,0,0,64,0],[51,44,0,0,0,0,67,22,0,0,0,0,0,0,0,0,64,8,0,0,0,0,8,9,0,0,8,9,0,0,0,0,9,65,0,0],[51,44,0,0,0,0,57,22,0,0,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0] >;`

C36.9D4 in GAP, Magma, Sage, TeX

`C_{36}._9D_4`
`% in TeX`

`G:=Group("C36.9D4");`
`// GroupNames label`

`G:=SmallGroup(288,42);`
`// by ID`

`G=gap.SmallGroup(288,42);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,28,141,120,219,100,675,6725,292,9414]);`
`// Polycyclic`

`G:=Group<a,b,c|a^36=1,b^4=a^18,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^17,c*b*c^-1=a^9*b^3>;`
`// generators/relations`

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