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## G = Dic6⋊D9order 432 = 24·33

### 2nd semidirect product of Dic6 and D9 acting via D9/C9=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C36 — Dic6⋊D9
 Chief series C1 — C3 — C9 — C3×C9 — C3×C18 — C3×C36 — C3×D36 — Dic6⋊D9
 Lower central C3×C9 — C3×C18 — C3×C36 — Dic6⋊D9
 Upper central C1 — C2 — C4

Generators and relations for Dic6⋊D9
G = < a,b,c,d | a12=c9=d2=1, b2=a6, bab-1=a-1, ac=ca, dad=a7, bc=cb, dbd=a3b, dcd=c-1 >

Subgroups: 376 in 68 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, Dic6, D12, C3×D4, C3×Q8, C3×C9, C36, C36, D18, C3×Dic3, C3×C12, S3×C6, D4.S3, Q82S3, C3×D9, C3×C18, C9⋊C8, D36, Q8×C9, C324C8, C3×Dic6, C3×D12, C9×Dic3, C3×C36, C6×D9, Q82D9, Dic6⋊S3, C36.S3, C9×Dic6, C3×D36, Dic6⋊D9
Quotients: C1, C2, C22, S3, D4, D6, SD16, D9, C3⋊D4, D18, S32, D4.S3, Q82S3, C9⋊D4, D6⋊S3, S3×D9, Q82D9, Dic6⋊S3, D6⋊D9, Dic6⋊D9

Smallest permutation representation of Dic6⋊D9
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 84 7 78)(2 83 8 77)(3 82 9 76)(4 81 10 75)(5 80 11 74)(6 79 12 73)(13 129 19 123)(14 128 20 122)(15 127 21 121)(16 126 22 132)(17 125 23 131)(18 124 24 130)(25 54 31 60)(26 53 32 59)(27 52 33 58)(28 51 34 57)(29 50 35 56)(30 49 36 55)(37 102 43 108)(38 101 44 107)(39 100 45 106)(40 99 46 105)(41 98 47 104)(42 97 48 103)(61 142 67 136)(62 141 68 135)(63 140 69 134)(64 139 70 133)(65 138 71 144)(66 137 72 143)(85 116 91 110)(86 115 92 109)(87 114 93 120)(88 113 94 119)(89 112 95 118)(90 111 96 117)
(1 59 21 5 51 13 9 55 17)(2 60 22 6 52 14 10 56 18)(3 49 23 7 53 15 11 57 19)(4 50 24 8 54 16 12 58 20)(25 132 79 33 128 75 29 124 83)(26 121 80 34 129 76 30 125 84)(27 122 81 35 130 77 31 126 73)(28 123 82 36 131 78 32 127 74)(37 71 120 41 63 112 45 67 116)(38 72 109 42 64 113 46 68 117)(39 61 110 43 65 114 47 69 118)(40 62 111 44 66 115 48 70 119)(85 108 138 93 104 134 89 100 142)(86 97 139 94 105 135 90 101 143)(87 98 140 95 106 136 91 102 144)(88 99 141 96 107 137 92 103 133)
(1 92)(2 87)(3 94)(4 89)(5 96)(6 91)(7 86)(8 93)(9 88)(10 95)(11 90)(12 85)(13 99)(14 106)(15 101)(16 108)(17 103)(18 98)(19 105)(20 100)(21 107)(22 102)(23 97)(24 104)(25 62)(26 69)(27 64)(28 71)(29 66)(30 61)(31 68)(32 63)(33 70)(34 65)(35 72)(36 67)(37 123)(38 130)(39 125)(40 132)(41 127)(42 122)(43 129)(44 124)(45 131)(46 126)(47 121)(48 128)(49 139)(50 134)(51 141)(52 136)(53 143)(54 138)(55 133)(56 140)(57 135)(58 142)(59 137)(60 144)(73 113)(74 120)(75 115)(76 110)(77 117)(78 112)(79 119)(80 114)(81 109)(82 116)(83 111)(84 118)

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

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

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

48 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 12F 18A 18B 18C 18D 18E 18F 36A ··· 36I 36J ··· 36O 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 12 18 18 18 18 18 18 36 ··· 36 36 ··· 36 size 1 1 36 2 2 4 2 12 2 2 4 36 36 54 54 2 2 2 4 4 4 4 4 4 4 12 12 2 2 2 4 4 4 4 ··· 4 12 ··· 12

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 SD16 D9 C3⋊D4 C3⋊D4 D18 C9⋊D4 S32 D4.S3 Q8⋊2S3 D6⋊S3 S3×D9 Q8⋊2D9 Dic6⋊S3 D6⋊D9 Dic6⋊D9 kernel Dic6⋊D9 C36.S3 C9×Dic6 C3×D36 D36 C3×Dic6 C3×C18 C36 C3×C12 C3×C9 Dic6 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 Dic6⋊D9 in GL6(𝔽73)

 1 71 0 0 0 0 1 72 0 0 0 0 0 0 0 72 0 0 0 0 1 72 0 0 0 0 0 0 72 0 0 0 0 0 0 72
,
 67 59 0 0 0 0 60 6 0 0 0 0 0 0 0 72 0 0 0 0 72 0 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 42 28 0 0 0 0 45 70
,
 11 52 0 0 0 0 37 62 0 0 0 0 0 0 72 0 0 0 0 0 0 72 0 0 0 0 0 0 10 41 0 0 0 0 51 63

G:=sub<GL(6,GF(73))| [1,1,0,0,0,0,71,72,0,0,0,0,0,0,0,1,0,0,0,0,72,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[67,60,0,0,0,0,59,6,0,0,0,0,0,0,0,72,0,0,0,0,72,0,0,0,0,0,0,0,30,60,0,0,0,0,13,43],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,42,45,0,0,0,0,28,70],[11,37,0,0,0,0,52,62,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,10,51,0,0,0,0,41,63] >;

Dic6⋊D9 in GAP, Magma, Sage, TeX

{\rm Dic}_6\rtimes D_9
% in TeX

G:=Group("Dic6:D9");
// GroupNames label

G:=SmallGroup(432,72);
// by ID

G=gap.SmallGroup(432,72);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,85,64,254,135,58,571,10085,292,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^12=c^9=d^2=1,b^2=a^6,b*a*b^-1=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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