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## G = C3⋊Dic36order 432 = 24·33

### The semidirect product of C3 and Dic36 acting via Dic36/Dic18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — C3⋊Dic36
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C3×Dic18 — C3⋊Dic36
 Lower central C3×C9 — C3×C18 — C3×C36 — C3⋊Dic36
 Upper central C1 — C2 — C4

Generators and relations for C3⋊Dic36
G = < a,b,c | a3=b72=1, c2=b36, bab-1=a-1, ac=ca, cbc-1=b-1 >

Subgroups: 400 in 66 conjugacy classes, 25 normal (all characteristic)
C1, C2, C3, C3, C4, C4, C6, C6, C8, Q8, C9, C9, C32, Dic3, C12, C12, Q16, C18, C18, C3×C6, C3⋊C8, C24, Dic6, C3×Q8, C3×C9, Dic9, C36, C36, C3×Dic3, C3⋊Dic3, C3×C12, Dic12, C3⋊Q16, C3×C18, C72, Dic18, Dic18, C3×C3⋊C8, C3×Dic6, C324Q8, C3×Dic9, C9⋊Dic3, C3×C36, Dic36, C323Q16, C9×C3⋊C8, C3×Dic18, C12.D9, C3⋊Dic36
Quotients: C1, C2, C22, S3, D4, D6, Q16, D9, D12, C3⋊D4, D18, S32, Dic12, C3⋊Q16, D36, C3⋊D12, S3×D9, Dic36, C323Q16, C3⋊D36, C3⋊Dic36

Smallest permutation representation of C3⋊Dic36
On 144 points
Generators in S144
(1 49 25)(2 26 50)(3 51 27)(4 28 52)(5 53 29)(6 30 54)(7 55 31)(8 32 56)(9 57 33)(10 34 58)(11 59 35)(12 36 60)(13 61 37)(14 38 62)(15 63 39)(16 40 64)(17 65 41)(18 42 66)(19 67 43)(20 44 68)(21 69 45)(22 46 70)(23 71 47)(24 48 72)(73 97 121)(74 122 98)(75 99 123)(76 124 100)(77 101 125)(78 126 102)(79 103 127)(80 128 104)(81 105 129)(82 130 106)(83 107 131)(84 132 108)(85 109 133)(86 134 110)(87 111 135)(88 136 112)(89 113 137)(90 138 114)(91 115 139)(92 140 116)(93 117 141)(94 142 118)(95 119 143)(96 144 120)
(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 99 37 135)(2 98 38 134)(3 97 39 133)(4 96 40 132)(5 95 41 131)(6 94 42 130)(7 93 43 129)(8 92 44 128)(9 91 45 127)(10 90 46 126)(11 89 47 125)(12 88 48 124)(13 87 49 123)(14 86 50 122)(15 85 51 121)(16 84 52 120)(17 83 53 119)(18 82 54 118)(19 81 55 117)(20 80 56 116)(21 79 57 115)(22 78 58 114)(23 77 59 113)(24 76 60 112)(25 75 61 111)(26 74 62 110)(27 73 63 109)(28 144 64 108)(29 143 65 107)(30 142 66 106)(31 141 67 105)(32 140 68 104)(33 139 69 103)(34 138 70 102)(35 137 71 101)(36 136 72 100)

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

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

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

60 conjugacy classes

 class 1 2 3A 3B 3C 4A 4B 4C 6A 6B 6C 8A 8B 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 18A 18B 18C 18D 18E 18F 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 72A ··· 72L order 1 2 3 3 3 4 4 4 6 6 6 8 8 9 9 9 9 9 9 12 12 12 12 12 12 12 18 18 18 18 18 18 24 24 24 24 36 ··· 36 36 ··· 36 72 ··· 72 size 1 1 2 2 4 2 36 108 2 2 4 6 6 2 2 2 4 4 4 2 2 4 4 4 36 36 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 Q16 D9 C3⋊D4 D12 D18 Dic12 D36 Dic36 S32 C3⋊Q16 C3⋊D12 S3×D9 C32⋊3Q16 C3⋊D36 C3⋊Dic36 kernel C3⋊Dic36 C9×C3⋊C8 C3×Dic18 C12.D9 Dic18 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 C3⋊Dic36 in GL6(𝔽73)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 72 72 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 50 18 0 0 0 0 55 68 0 0 0 0 0 0 1 0 0 0 0 0 72 72 0 0 0 0 0 0 29 19 0 0 0 0 54 48
,
 18 53 0 0 0 0 71 55 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 1 0 0 0 0 0 1 72

G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[50,55,0,0,0,0,18,68,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,29,54,0,0,0,0,19,48],[18,71,0,0,0,0,53,55,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,1,0,0,0,0,0,72] >;

C3⋊Dic36 in GAP, Magma, Sage, TeX

C_3\rtimes {\rm Dic}_{36}
% in TeX

G:=Group("C3:Dic36");
// GroupNames label

G:=SmallGroup(432,65);
// by ID

G=gap.SmallGroup(432,65);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,85,92,254,58,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c|a^3=b^72=1,c^2=b^36,b*a*b^-1=a^-1,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations

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