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## G = Dic3⋊Dic9order 432 = 24·33

### The semidirect product of Dic3 and Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — Dic3⋊Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C6×C18 — Dic3×C18 — Dic3⋊Dic9
 Lower central C3×C9 — C3×C18 — Dic3⋊Dic9
 Upper central C1 — C22

Generators and relations for Dic3⋊Dic9
G = < a,b,c,d | a6=c18=1, b2=a3, d2=c9, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c-1 >

Subgroups: 444 in 94 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C3, C3, C4, C22, C6, C6, C2×C4, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C18, C18, C3×C6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, Dic3⋊C4, C4⋊Dic3, C3×C18, C2×Dic9, C2×Dic9, C2×C36, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C4⋊Dic9, Dic3⋊Dic3, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, Dic3⋊Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C4⋊C4, D9, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, Dic9, D18, S32, Dic3⋊C4, C4⋊Dic3, Dic18, D36, C2×Dic9, S3×Dic3, C3⋊D12, C322Q8, S3×D9, C4⋊Dic9, Dic3⋊Dic3, C9⋊Dic6, C3⋊D36, S3×Dic9, Dic3⋊Dic9

Smallest permutation representation of Dic3⋊Dic9
On 144 points
Generators in S144
(1 48 7 54 13 42)(2 49 8 37 14 43)(3 50 9 38 15 44)(4 51 10 39 16 45)(5 52 11 40 17 46)(6 53 12 41 18 47)(19 56 31 68 25 62)(20 57 32 69 26 63)(21 58 33 70 27 64)(22 59 34 71 28 65)(23 60 35 72 29 66)(24 61 36 55 30 67)(73 105 85 99 79 93)(74 106 86 100 80 94)(75 107 87 101 81 95)(76 108 88 102 82 96)(77 91 89 103 83 97)(78 92 90 104 84 98)(109 133 115 139 121 127)(110 134 116 140 122 128)(111 135 117 141 123 129)(112 136 118 142 124 130)(113 137 119 143 125 131)(114 138 120 144 126 132)
(1 80 54 106)(2 81 37 107)(3 82 38 108)(4 83 39 91)(5 84 40 92)(6 85 41 93)(7 86 42 94)(8 87 43 95)(9 88 44 96)(10 89 45 97)(11 90 46 98)(12 73 47 99)(13 74 48 100)(14 75 49 101)(15 76 50 102)(16 77 51 103)(17 78 52 104)(18 79 53 105)(19 133 68 121)(20 134 69 122)(21 135 70 123)(22 136 71 124)(23 137 72 125)(24 138 55 126)(25 139 56 109)(26 140 57 110)(27 141 58 111)(28 142 59 112)(29 143 60 113)(30 144 61 114)(31 127 62 115)(32 128 63 116)(33 129 64 117)(34 130 65 118)(35 131 66 119)(36 132 67 120)
(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 64 10 55)(2 63 11 72)(3 62 12 71)(4 61 13 70)(5 60 14 69)(6 59 15 68)(7 58 16 67)(8 57 17 66)(9 56 18 65)(19 41 28 50)(20 40 29 49)(21 39 30 48)(22 38 31 47)(23 37 32 46)(24 54 33 45)(25 53 34 44)(26 52 35 43)(27 51 36 42)(73 136 82 127)(74 135 83 144)(75 134 84 143)(76 133 85 142)(77 132 86 141)(78 131 87 140)(79 130 88 139)(80 129 89 138)(81 128 90 137)(91 114 100 123)(92 113 101 122)(93 112 102 121)(94 111 103 120)(95 110 104 119)(96 109 105 118)(97 126 106 117)(98 125 107 116)(99 124 108 115)

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

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

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

66 conjugacy classes

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

66 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 type + + + + + + + - + - + + - - + - + - + + - + - + - + - image C1 C2 C2 C2 C4 S3 S3 D4 Q8 D6 Dic3 D6 D9 Dic6 C4×S3 C3⋊D4 Dic6 D12 Dic9 D18 Dic18 D36 S32 S3×Dic3 C3⋊D12 C32⋊2Q8 S3×D9 C9⋊Dic6 C3⋊D36 S3×Dic9 kernel Dic3⋊Dic9 C6×Dic9 Dic3×C18 C2×C9⋊Dic3 C9×Dic3 C2×Dic9 C6×Dic3 C3×C18 C3×C18 C2×C18 C3×Dic3 C62 C2×Dic3 C18 C18 C18 C3×C6 C3×C6 Dic3 C2×C6 C6 C6 C2×C6 C6 C6 C6 C22 C2 C2 C2 # reps 1 1 1 1 4 1 1 1 1 1 2 1 3 2 2 2 2 2 6 3 6 6 1 1 1 1 3 3 3 3

Matrix representation of Dic3⋊Dic9 in GL6(𝔽37)

 36 0 0 0 0 0 0 36 0 0 0 0 0 0 0 36 0 0 0 0 1 36 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 5 27 0 0 0 0 10 32 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 0 1
,
 36 1 0 0 0 0 36 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 31 26 0 0 0 0 11 20
,
 31 17 0 0 0 0 11 6 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 31 6 0 0 0 0 0 6

G:=sub<GL(6,GF(37))| [36,0,0,0,0,0,0,36,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[5,10,0,0,0,0,27,32,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[36,36,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,11,0,0,0,0,26,20],[31,11,0,0,0,0,17,6,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,31,0,0,0,0,0,6,6] >;

Dic3⋊Dic9 in GAP, Magma, Sage, TeX

{\rm Dic}_3\rtimes {\rm Dic}_9
% in TeX

G:=Group("Dic3:Dic9");
// GroupNames label

G:=SmallGroup(432,90);
// by ID

G=gap.SmallGroup(432,90);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,36,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^6=c^18=1,b^2=a^3,d^2=c^9,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=c^-1>;
// generators/relations

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