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## G = C36.49D4order 288 = 25·32

### 5th non-split extension by C36 of D4 acting via D4/C22=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.49D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C2×Dic9 — C2×Dic18 — C36.49D4
 Lower central C9 — C2×C18 — C36.49D4
 Upper central C1 — C22 — C22×C4

Generators and relations for C36.49D4
G = < a,b,c | a36=b4=1, c2=a18, bab-1=cac-1=a-1, cbc-1=a18b-1 >

Subgroups: 380 in 111 conjugacy classes, 50 normal (30 characteristic)
C1, C2 [×3], C2 [×2], C3, C4 [×2], C4 [×5], C22, C22 [×2], C22 [×2], C6 [×3], C6 [×2], C2×C4 [×2], C2×C4 [×6], Q8 [×2], C23, C9, Dic3 [×4], C12 [×2], C12, C2×C6, C2×C6 [×2], C2×C6 [×2], C22⋊C4 [×2], C4⋊C4 [×3], C22×C4, C2×Q8, C18 [×3], C18 [×2], Dic6 [×2], C2×Dic3 [×4], C2×C12 [×2], C2×C12 [×2], C22×C6, C22⋊Q8, Dic9 [×4], C36 [×2], C36, C2×C18, C2×C18 [×2], C2×C18 [×2], Dic3⋊C4 [×2], C4⋊Dic3, C6.D4 [×2], C2×Dic6, C22×C12, Dic18 [×2], C2×Dic9 [×4], C2×C36 [×2], C2×C36 [×2], C22×C18, C12.48D4, Dic9⋊C4 [×2], C4⋊Dic9, C18.D4 [×2], C2×Dic18, C22×C36, C36.49D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], Q8 [×2], C23, D6 [×3], C2×D4, C2×Q8, C4○D4, D9, Dic6 [×2], C3⋊D4 [×2], C22×S3, C22⋊Q8, D18 [×3], C2×Dic6, C4○D12, C2×C3⋊D4, Dic18 [×2], C9⋊D4 [×2], C22×D9, C12.48D4, C2×Dic18, D365C2, C2×C9⋊D4, C36.49D4

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

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

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

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

78 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 4D 4E 4F 4G 4H 6A ··· 6G 9A 9B 9C 12A ··· 12H 18A ··· 18U 36A ··· 36X order 1 2 2 2 2 2 3 4 4 4 4 4 4 4 4 6 ··· 6 9 9 9 12 ··· 12 18 ··· 18 36 ··· 36 size 1 1 1 1 2 2 2 2 2 2 2 36 36 36 36 2 ··· 2 2 2 2 2 ··· 2 2 ··· 2 2 ··· 2

78 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + - + + + - + + - image C1 C2 C2 C2 C2 C2 S3 D4 Q8 D6 D6 C4○D4 D9 C3⋊D4 Dic6 D18 D18 C4○D12 C9⋊D4 Dic18 D36⋊5C2 kernel C36.49D4 Dic9⋊C4 C4⋊Dic9 C18.D4 C2×Dic18 C22×C36 C22×C12 C36 C2×C18 C2×C12 C22×C6 C18 C22×C4 C12 C2×C6 C2×C4 C23 C6 C4 C22 C2 # reps 1 2 1 2 1 1 1 2 2 2 1 2 3 4 4 6 3 4 12 12 12

Matrix representation of C36.49D4 in GL4(𝔽37) generated by

 8 35 0 0 0 14 0 0 0 0 13 0 0 0 0 20
,
 36 0 0 0 3 1 0 0 0 0 0 1 0 0 36 0
,
 1 13 0 0 34 36 0 0 0 0 0 1 0 0 36 0
`G:=sub<GL(4,GF(37))| [8,0,0,0,35,14,0,0,0,0,13,0,0,0,0,20],[36,3,0,0,0,1,0,0,0,0,0,36,0,0,1,0],[1,34,0,0,13,36,0,0,0,0,0,36,0,0,1,0] >;`

C36.49D4 in GAP, Magma, Sage, TeX

`C_{36}._{49}D_4`
`% in TeX`

`G:=Group("C36.49D4");`
`// GroupNames label`

`G:=SmallGroup(288,134);`
`// by ID`

`G=gap.SmallGroup(288,134);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,112,253,120,254,6725,292,9414]);`
`// Polycyclic`

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

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