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## G = C4.D36order 288 = 25·32

### 3rd non-split extension by C4 of D36 acting via D36/D18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C4.D36
 Chief series C1 — C3 — C9 — C18 — C36 — C2×C36 — C2×Dic18 — C4.D36
 Lower central C9 — C18 — C2×C18 — C4.D36
 Upper central C1 — C2 — C2×C4 — M4(2)

Generators and relations for C4.D36
G = < a,b,c | a36=1, b4=c2=a18, bab-1=cac-1=a-1, cbc-1=a9b3 >

Subgroups: 228 in 57 conjugacy classes, 26 normal (24 characteristic)
C1, C2, C2, C3, C4 [×2], C4 [×2], C22, C6, C6, C8 [×2], C2×C4, C2×C4 [×2], Q8 [×2], C9, Dic3 [×2], C12 [×2], C2×C6, M4(2), M4(2), C2×Q8, C18, C18, C3⋊C8, C24, Dic6 [×2], C2×Dic3 [×2], C2×C12, C4.10D4, Dic9 [×2], C36 [×2], C2×C18, C4.Dic3, C3×M4(2), C2×Dic6, C9⋊C8, C72, Dic18 [×2], C2×Dic9 [×2], C2×C36, C12.47D4, C4.Dic9, C9×M4(2), C2×Dic18, C4.D36
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D4 [×2], D6, C22⋊C4, D9, C4×S3, D12, C3⋊D4, C4.10D4, D18, D6⋊C4, C4×D9, D36, C9⋊D4, C12.47D4, D18⋊C4, C4.D36

Smallest permutation representation of C4.D36
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)
(1 127 10 118 19 109 28 136)(2 126 11 117 20 144 29 135)(3 125 12 116 21 143 30 134)(4 124 13 115 22 142 31 133)(5 123 14 114 23 141 32 132)(6 122 15 113 24 140 33 131)(7 121 16 112 25 139 34 130)(8 120 17 111 26 138 35 129)(9 119 18 110 27 137 36 128)(37 90 64 99 55 108 46 81)(38 89 65 98 56 107 47 80)(39 88 66 97 57 106 48 79)(40 87 67 96 58 105 49 78)(41 86 68 95 59 104 50 77)(42 85 69 94 60 103 51 76)(43 84 70 93 61 102 52 75)(44 83 71 92 62 101 53 74)(45 82 72 91 63 100 54 73)
(1 90 19 108)(2 89 20 107)(3 88 21 106)(4 87 22 105)(5 86 23 104)(6 85 24 103)(7 84 25 102)(8 83 26 101)(9 82 27 100)(10 81 28 99)(11 80 29 98)(12 79 30 97)(13 78 31 96)(14 77 32 95)(15 76 33 94)(16 75 34 93)(17 74 35 92)(18 73 36 91)(37 118 55 136)(38 117 56 135)(39 116 57 134)(40 115 58 133)(41 114 59 132)(42 113 60 131)(43 112 61 130)(44 111 62 129)(45 110 63 128)(46 109 64 127)(47 144 65 126)(48 143 66 125)(49 142 67 124)(50 141 68 123)(51 140 69 122)(52 139 70 121)(53 138 71 120)(54 137 72 119)```

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

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

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

51 conjugacy classes

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

51 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + - - - image C1 C2 C2 C2 C4 S3 D4 D6 D9 D12 C3⋊D4 C4×S3 D18 D36 C9⋊D4 C4×D9 C4.10D4 C12.47D4 C4.D36 kernel C4.D36 C4.Dic9 C9×M4(2) C2×Dic18 C2×Dic9 C3×M4(2) C36 C2×C12 M4(2) C12 C12 C2×C6 C2×C4 C4 C4 C22 C9 C3 C1 # reps 1 1 1 1 4 1 2 1 3 2 2 2 3 6 6 6 1 2 6

Matrix representation of C4.D36 in GL4(𝔽73) generated by

 25 19 0 0 54 44 0 0 0 0 25 19 0 0 54 44
,
 0 0 14 5 0 0 19 59 51 12 0 0 63 22 0 0
,
 14 5 0 0 19 59 0 0 0 0 14 5 0 0 19 59
`G:=sub<GL(4,GF(73))| [25,54,0,0,19,44,0,0,0,0,25,54,0,0,19,44],[0,0,51,63,0,0,12,22,14,19,0,0,5,59,0,0],[14,19,0,0,5,59,0,0,0,0,14,19,0,0,5,59] >;`

C4.D36 in GAP, Magma, Sage, TeX

`C_4.D_{36}`
`% in TeX`

`G:=Group("C4.D36");`
`// GroupNames label`

`G:=SmallGroup(288,30);`
`// by ID`

`G=gap.SmallGroup(288,30);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,141,36,422,100,346,6725,292,9414]);`
`// Polycyclic`

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

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