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## G = C42⋊6D9order 288 = 25·32

### 5th semidirect product of C42 and D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C42⋊6D9
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×D36 — C42⋊6D9
 Lower central C9 — C2×C18 — C42⋊6D9
 Upper central C1 — C22 — C42

Generators and relations for C426D9
G = < a,b,c | a36=b4=c2=1, ab=ba, cac=a-1, cbc=b-1 >

Subgroups: 1020 in 162 conjugacy classes, 56 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, S3, C6, C2×C4, D4, C23, C9, C12, D6, C2×C6, C42, C2×D4, D9, C18, D12, C2×C12, C22×S3, C41D4, C36, D18, C2×C18, C4×C12, C2×D12, D36, C2×C36, C22×D9, C4⋊D12, C4×C36, C2×D36, C426D9
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D9, D12, C22×S3, C41D4, D18, C2×D12, D36, C22×D9, C4⋊D12, C2×D36, C426D9

Smallest permutation representation of C426D9
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 93 48 118)(2 94 49 119)(3 95 50 120)(4 96 51 121)(5 97 52 122)(6 98 53 123)(7 99 54 124)(8 100 55 125)(9 101 56 126)(10 102 57 127)(11 103 58 128)(12 104 59 129)(13 105 60 130)(14 106 61 131)(15 107 62 132)(16 108 63 133)(17 73 64 134)(18 74 65 135)(19 75 66 136)(20 76 67 137)(21 77 68 138)(22 78 69 139)(23 79 70 140)(24 80 71 141)(25 81 72 142)(26 82 37 143)(27 83 38 144)(28 84 39 109)(29 85 40 110)(30 86 41 111)(31 87 42 112)(32 88 43 113)(33 89 44 114)(34 90 45 115)(35 91 46 116)(36 92 47 117)
(1 48)(2 47)(3 46)(4 45)(5 44)(6 43)(7 42)(8 41)(9 40)(10 39)(11 38)(12 37)(13 72)(14 71)(15 70)(16 69)(17 68)(18 67)(19 66)(20 65)(21 64)(22 63)(23 62)(24 61)(25 60)(26 59)(27 58)(28 57)(29 56)(30 55)(31 54)(32 53)(33 52)(34 51)(35 50)(36 49)(73 77)(74 76)(78 108)(79 107)(80 106)(81 105)(82 104)(83 103)(84 102)(85 101)(86 100)(87 99)(88 98)(89 97)(90 96)(91 95)(92 94)(109 127)(110 126)(111 125)(112 124)(113 123)(114 122)(115 121)(116 120)(117 119)(128 144)(129 143)(130 142)(131 141)(132 140)(133 139)(134 138)(135 137)```

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

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

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

78 conjugacy classes

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

78 irreducible representations

 dim 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 S3 D4 D6 D9 D12 D18 D36 kernel C42⋊6D9 C4×C36 C2×D36 C4×C12 C36 C2×C12 C42 C12 C2×C4 C4 # reps 1 1 6 1 6 3 3 12 9 36

Matrix representation of C426D9 in GL6(𝔽37)

 27 32 0 0 0 0 5 32 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 26 17 0 0 0 0 20 6
,
 5 10 0 0 0 0 27 32 0 0 0 0 0 0 15 3 0 0 0 0 11 22 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 36 0 0 0 0 0 1 1 0 0 0 0 0 0 36 0 0 0 0 0 10 1 0 0 0 0 0 0 0 1 0 0 0 0 1 0

`G:=sub<GL(6,GF(37))| [27,5,0,0,0,0,32,32,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,26,20,0,0,0,0,17,6],[5,27,0,0,0,0,10,32,0,0,0,0,0,0,15,11,0,0,0,0,3,22,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,1,0,0,0,0,0,1,0,0,0,0,0,0,36,10,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;`

C426D9 in GAP, Magma, Sage, TeX

`C_4^2\rtimes_6D_9`
`% in TeX`

`G:=Group("C4^2:6D9");`
`// GroupNames label`

`G:=SmallGroup(288,84);`
`// by ID`

`G=gap.SmallGroup(288,84);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,58,6725,292,9414]);`
`// Polycyclic`

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

׿
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