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## G = D36⋊S3order 432 = 24·33

### 2nd semidirect product of D36 and S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — D36⋊S3
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — C9×D12 — D36⋊S3
 Lower central C3×C9 — C3×C18 — C3×C36 — D36⋊S3
 Upper central C1 — C2 — C4

Generators and relations for D36⋊S3
G = < a,b,c,d | a12=b2=c9=d2=1, bab=a-1, ac=ca, dad=a7, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 424 in 74 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C22, S3, C6, C6, C8, D4, C9, C9, C32, C12, C12, D6, C2×C6, D8, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, D12, D12, C3×D4, C3×C9, C36, C36, D18, C2×C18, C3×C12, S3×C6, D4⋊S3, C3×D9, S3×C9, C3×C18, C9⋊C8, D36, D4×C9, C324C8, C3×D12, C3×D12, C3×C36, C6×D9, S3×C18, D4⋊D9, C322D8, C36.S3, C3×D36, C9×D12, D36⋊S3
Quotients: C1, C2, C22, S3, D4, D6, D8, D9, C3⋊D4, D18, S32, D4⋊S3, C9⋊D4, D6⋊S3, S3×D9, D4⋊D9, C322D8, D6⋊D9, D36⋊S3

Smallest permutation representation of D36⋊S3
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)
(1 143)(2 142)(3 141)(4 140)(5 139)(6 138)(7 137)(8 136)(9 135)(10 134)(11 133)(12 144)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 48)(20 47)(21 46)(22 45)(23 44)(24 43)(25 116)(26 115)(27 114)(28 113)(29 112)(30 111)(31 110)(32 109)(33 120)(34 119)(35 118)(36 117)(49 127)(50 126)(51 125)(52 124)(53 123)(54 122)(55 121)(56 132)(57 131)(58 130)(59 129)(60 128)(61 99)(62 98)(63 97)(64 108)(65 107)(66 106)(67 105)(68 104)(69 103)(70 102)(71 101)(72 100)(73 85)(74 96)(75 95)(76 94)(77 93)(78 92)(79 91)(80 90)(81 89)(82 88)(83 87)(84 86)
(1 130 91 9 126 87 5 122 95)(2 131 92 10 127 88 6 123 96)(3 132 93 11 128 89 7 124 85)(4 121 94 12 129 90 8 125 86)(13 117 62 21 113 70 17 109 66)(14 118 63 22 114 71 18 110 67)(15 119 64 23 115 72 19 111 68)(16 120 65 24 116 61 20 112 69)(25 99 47 29 103 39 33 107 43)(26 100 48 30 104 40 34 108 44)(27 101 37 31 105 41 35 97 45)(28 102 38 32 106 42 36 98 46)(49 82 138 53 74 142 57 78 134)(50 83 139 54 75 143 58 79 135)(51 84 140 55 76 144 59 80 136)(52 73 141 56 77 133 60 81 137)
(1 33)(2 28)(3 35)(4 30)(5 25)(6 32)(7 27)(8 34)(9 29)(10 36)(11 31)(12 26)(13 58)(14 53)(15 60)(16 55)(17 50)(18 57)(19 52)(20 59)(21 54)(22 49)(23 56)(24 51)(37 128)(38 123)(39 130)(40 125)(41 132)(42 127)(43 122)(44 129)(45 124)(46 131)(47 126)(48 121)(61 80)(62 75)(63 82)(64 77)(65 84)(66 79)(67 74)(68 81)(69 76)(70 83)(71 78)(72 73)(85 97)(86 104)(87 99)(88 106)(89 101)(90 108)(91 103)(92 98)(93 105)(94 100)(95 107)(96 102)(109 135)(110 142)(111 137)(112 144)(113 139)(114 134)(115 141)(116 136)(117 143)(118 138)(119 133)(120 140)```

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

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

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

48 conjugacy classes

 class 1 2A 2B 2C 3A 3B 3C 4 6A 6B 6C 6D 6E 6F 6G 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 18A 18B 18C 18D 18E 18F 18G ··· 18L 36A ··· 36I order 1 2 2 2 3 3 3 4 6 6 6 6 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 18 18 18 18 18 18 18 ··· 18 36 ··· 36 size 1 1 12 36 2 2 4 2 2 2 4 12 12 36 36 54 54 2 2 2 4 4 4 4 4 4 4 2 2 2 4 4 4 12 ··· 12 4 ··· 4

48 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + + - + + - image C1 C2 C2 C2 S3 S3 D4 D6 D6 D8 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 S32 D4⋊S3 D4⋊S3 D6⋊S3 S3×D9 D4⋊D9 C32⋊2D8 D6⋊D9 D36⋊S3 kernel D36⋊S3 C36.S3 C3×D36 C9×D12 D36 C3×D12 C3×C18 C36 C3×C12 C3×C9 D12 C18 C3×C6 C12 C6 C12 C9 C32 C6 C4 C3 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 3 2 2 3 6 1 1 1 1 3 3 2 3 6

Matrix representation of D36⋊S3 in GL6(𝔽73)

 0 1 0 0 0 0 72 0 0 0 0 0 0 0 72 1 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 7 68 0 0 0 0 68 66 0 0 0 0 0 0 72 0 0 0 0 0 72 1 0 0 0 0 0 0 30 60 0 0 0 0 13 43
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 3 0 0 0 0 70 28
,
 32 27 0 0 0 0 27 41 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 0 72 0 0 0 0 72 0

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[7,68,0,0,0,0,68,66,0,0,0,0,0,0,72,72,0,0,0,0,0,1,0,0,0,0,0,0,30,13,0,0,0,0,60,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,70,0,0,0,0,3,28],[32,27,0,0,0,0,27,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,72,0] >;`

D36⋊S3 in GAP, Magma, Sage, TeX

`D_{36}\rtimes S_3`
`% in TeX`

`G:=Group("D36:S3");`
`// GroupNames label`

`G:=SmallGroup(432,68);`
`// by ID`

`G=gap.SmallGroup(432,68);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,254,135,58,3091,662,4037,7069]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^12=b^2=c^9=d^2=1,b*a*b=a^-1,a*c=c*a,d*a*d=a^7,b*c=c*b,d*b*d=a^3*b,d*c*d=c^-1>;`
`// generators/relations`

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