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## G = D36.S3order 432 = 24·33

### 1st non-split extension by D36 of S3 acting via S3/C3=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — D36.S3
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C3×D36 — 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 | a36=b2=c3=1, d2=a9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a27b, dcd-1=c-1 >

Subgroups: 504 in 72 conjugacy classes, 25 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, SD16, D9, C18, C18, C3×S3, C3×C6, C3⋊C8, C24, Dic6, D12, C3×D4, C3×C9, Dic9, C36, C36, D18, C3⋊Dic3, C3×C12, S3×C6, C24⋊C2, D4.S3, C3×D9, C3×C18, C72, Dic18, D36, C3×C3⋊C8, C3×D12, C324Q8, C9⋊Dic3, C3×C36, C6×D9, C72⋊C2, D12.S3, C9×C3⋊C8, C3×D36, C12.D9, D36.S3
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, D12, C3⋊D4, D18, S32, C24⋊C2, D4.S3, D36, C3⋊D12, S3×D9, C72⋊C2, D12.S3, C3⋊D36, 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 78)(2 77)(3 76)(4 75)(5 74)(6 73)(7 108)(8 107)(9 106)(10 105)(11 104)(12 103)(13 102)(14 101)(15 100)(16 99)(17 98)(18 97)(19 96)(20 95)(21 94)(22 93)(23 92)(24 91)(25 90)(26 89)(27 88)(28 87)(29 86)(30 85)(31 84)(32 83)(33 82)(34 81)(35 80)(36 79)(37 126)(38 125)(39 124)(40 123)(41 122)(42 121)(43 120)(44 119)(45 118)(46 117)(47 116)(48 115)(49 114)(50 113)(51 112)(52 111)(53 110)(54 109)(55 144)(56 143)(57 142)(58 141)(59 140)(60 139)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 132)(68 131)(69 130)(70 129)(71 128)(72 127)
(1 13 25)(2 14 26)(3 15 27)(4 16 28)(5 17 29)(6 18 30)(7 19 31)(8 20 32)(9 21 33)(10 22 34)(11 23 35)(12 24 36)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 121 133)(110 122 134)(111 123 135)(112 124 136)(113 125 137)(114 126 138)(115 127 139)(116 128 140)(117 129 141)(118 130 142)(119 131 143)(120 132 144)
(1 38 10 47 19 56 28 65)(2 39 11 48 20 57 29 66)(3 40 12 49 21 58 30 67)(4 41 13 50 22 59 31 68)(5 42 14 51 23 60 32 69)(6 43 15 52 24 61 33 70)(7 44 16 53 25 62 34 71)(8 45 17 54 26 63 35 72)(9 46 18 55 27 64 36 37)(73 111 82 120 91 129 100 138)(74 112 83 121 92 130 101 139)(75 113 84 122 93 131 102 140)(76 114 85 123 94 132 103 141)(77 115 86 124 95 133 104 142)(78 116 87 125 96 134 105 143)(79 117 88 126 97 135 106 144)(80 118 89 127 98 136 107 109)(81 119 90 128 99 137 108 110)```

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

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

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

60 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 6A 6B 6C 6D 6E 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 2 3 3 3 4 4 6 6 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 36 2 2 4 2 108 2 2 4 36 36 6 6 2 2 2 4 4 4 2 2 4 4 4 2 2 2 4 4 4 6 6 6 6 2 ··· 2 4 ··· 4 6 ··· 6

60 irreducible representations

 dim 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 type + + + + + + + + + + + + + + - + + - + - image C1 C2 C2 C2 S3 S3 D4 D6 D6 SD16 D9 C3⋊D4 D12 D18 C24⋊C2 D36 C72⋊C2 S32 D4.S3 C3⋊D12 S3×D9 D12.S3 C3⋊D36 D36.S3 kernel D36.S3 C9×C3⋊C8 C3×D36 C12.D9 D36 C3×C3⋊C8 C3×C18 C36 C3×C12 C3×C9 C3⋊C8 C18 C3×C6 C12 C32 C6 C3 C12 C9 C6 C4 C3 C2 C1 # reps 1 1 1 1 1 1 1 1 1 2 3 2 2 3 4 6 12 1 1 1 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 28 0 0 0 0 45 3
,
 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 18 68 0 0 0 0 50 55
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 72 72 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 67 6 0 0 0 0 67 67 0 0 0 0 0 0 72 0 0 0 0 0 1 1 0 0 0 0 0 0 66 59 0 0 0 0 14 7

`G:=sub<GL(6,GF(73))| [0,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,45,0,0,0,0,28,3],[32,27,0,0,0,0,27,41,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,18,50,0,0,0,0,68,55],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[67,67,0,0,0,0,6,67,0,0,0,0,0,0,72,1,0,0,0,0,0,1,0,0,0,0,0,0,66,14,0,0,0,0,59,7] >;`

D36.S3 in GAP, Magma, Sage, TeX

`D_{36}.S_3`
`% in TeX`

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

`G:=SmallGroup(432,62);`
`// by ID`

`G=gap.SmallGroup(432,62);`
`# by ID`

`G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,36,254,58,571,10085,292,14118]);`
`// Polycyclic`

`G:=Group<a,b,c,d|a^36=b^2=c^3=1,d^2=a^9,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^27*b,d*c*d^-1=c^-1>;`
`// generators/relations`

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