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## G = C36.23D4order 288 = 25·32

### 23rd non-split extension by C36 of D4 acting via D4/C2=C22

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C18 — C36.23D4
 Chief series C1 — C3 — C9 — C18 — C2×C18 — C22×D9 — C2×D36 — C36.23D4
 Lower central C9 — C2×C18 — C36.23D4
 Upper central C1 — C22 — C2×Q8

Generators and relations for C36.23D4
G = < a,b,c | a36=b4=c2=1, bab-1=a17, cac=a-1, cbc=a18b-1 >

Subgroups: 564 in 114 conjugacy classes, 44 normal (18 characteristic)
C1, C2, C2 [×2], C2 [×2], C3, C4 [×2], C4 [×4], C22, C22 [×6], S3 [×2], C6, C6 [×2], C2×C4, C2×C4 [×2], C2×C4 [×2], D4 [×2], Q8 [×2], C23 [×2], C9, Dic3 [×2], C12 [×2], C12 [×2], D6 [×6], C2×C6, C42, C22⋊C4 [×4], C2×D4, C2×Q8, D9 [×2], C18, C18 [×2], D12 [×2], C2×Dic3 [×2], C2×C12, C2×C12 [×2], C3×Q8 [×2], C22×S3 [×2], C4.4D4, Dic9 [×2], C36 [×2], C36 [×2], D18 [×6], C2×C18, C4×Dic3, D6⋊C4 [×4], C2×D12, C6×Q8, D36 [×2], C2×Dic9 [×2], C2×C36, C2×C36 [×2], Q8×C9 [×2], C22×D9 [×2], C12.23D4, C4×Dic9, D18⋊C4 [×4], C2×D36, Q8×C18, C36.23D4
Quotients: C1, C2 [×7], C22 [×7], S3, D4 [×2], C23, D6 [×3], C2×D4, C4○D4 [×2], D9, C3⋊D4 [×2], C22×S3, C4.4D4, D18 [×3], Q83S3 [×2], C2×C3⋊D4, C9⋊D4 [×2], C22×D9, C12.23D4, Q83D9 [×2], C2×C9⋊D4, C36.23D4

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

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 type + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D4 D6 C4○D4 D9 C3⋊D4 D18 C9⋊D4 Q8⋊3S3 Q8⋊3D9 kernel C36.23D4 C4×Dic9 D18⋊C4 C2×D36 Q8×C18 C6×Q8 C36 C2×C12 C18 C2×Q8 C12 C2×C4 C4 C6 C2 # reps 1 1 4 1 1 1 2 3 4 3 4 9 12 2 6

Matrix representation of C36.23D4 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 6 11 0 0 0 0 26 17 0 0 0 0 0 0 1 4 0 0 0 0 18 36
,
 21 6 0 0 0 0 25 16 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 31 13 0 0 0 0 3 6
,
 1 0 0 0 0 0 30 36 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 18 36

`G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,6,26,0,0,0,0,11,17,0,0,0,0,0,0,1,18,0,0,0,0,4,36],[21,25,0,0,0,0,6,16,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,31,3,0,0,0,0,13,6],[1,30,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,18,0,0,0,0,0,36] >;`

C36.23D4 in GAP, Magma, Sage, TeX

`C_{36}._{23}D_4`
`% in TeX`

`G:=Group("C36.23D4");`
`// GroupNames label`

`G:=SmallGroup(288,157);`
`// by ID`

`G=gap.SmallGroup(288,157);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,254,219,100,6725,292,9414]);`
`// Polycyclic`

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

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