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## G = D9×Dic6order 432 = 24·33

### Direct product of D9 and Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D9×Dic6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — Dic3×D9 — D9×Dic6
 Lower central C3×C9 — C3×C18 — D9×Dic6
 Upper central C1 — C2 — C4

Generators and relations for D9×Dic6
G = < a,b,c,d | a9=b2=c12=1, d2=c6, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 684 in 126 conjugacy classes, 45 normal (29 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C12, D6, C2×C6, C2×Q8, D9, C18, C18, C3×S3, C3×C6, Dic6, Dic6, C4×S3, C2×Dic3, C2×C12, C3×Q8, C3×C9, Dic9, Dic9, C36, C36, D18, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, C2×Dic6, S3×Q8, C3×D9, C3×C18, Dic18, C4×D9, C4×D9, Q8×C9, S3×Dic3, C322Q8, C3×Dic6, S3×C12, C324Q8, C3×Dic9, C9×Dic3, C9⋊Dic3, C3×C36, C6×D9, Q8×D9, S3×Dic6, C9⋊Dic6, Dic3×D9, C9×Dic6, C12×D9, C12.D9, D9×Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, S3×Q8, C22×D9, C2×S32, S3×D9, Q8×D9, S3×Dic6, C2×S3×D9, D9×Dic6

Smallest permutation representation of D9×Dic6
On 144 points
Generators in S144
(1 40 67 5 44 71 9 48 63)(2 41 68 6 45 72 10 37 64)(3 42 69 7 46 61 11 38 65)(4 43 70 8 47 62 12 39 66)(13 50 30 21 58 26 17 54 34)(14 51 31 22 59 27 18 55 35)(15 52 32 23 60 28 19 56 36)(16 53 33 24 49 29 20 57 25)(73 132 96 77 124 88 81 128 92)(74 121 85 78 125 89 82 129 93)(75 122 86 79 126 90 83 130 94)(76 123 87 80 127 91 84 131 95)(97 113 140 105 109 136 101 117 144)(98 114 141 106 110 137 102 118 133)(99 115 142 107 111 138 103 119 134)(100 116 143 108 112 139 104 120 135)
(1 138)(2 139)(3 140)(4 141)(5 142)(6 143)(7 144)(8 133)(9 134)(10 135)(11 136)(12 137)(13 84)(14 73)(15 74)(16 75)(17 76)(18 77)(19 78)(20 79)(21 80)(22 81)(23 82)(24 83)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 121)(37 120)(38 109)(39 110)(40 111)(41 112)(42 113)(43 114)(44 115)(45 116)(46 117)(47 118)(48 119)(49 90)(50 91)(51 92)(52 93)(53 94)(54 95)(55 96)(56 85)(57 86)(58 87)(59 88)(60 89)(61 101)(62 102)(63 103)(64 104)(65 105)(66 106)(67 107)(68 108)(69 97)(70 98)(71 99)(72 100)
(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 24 7 18)(2 23 8 17)(3 22 9 16)(4 21 10 15)(5 20 11 14)(6 19 12 13)(25 65 31 71)(26 64 32 70)(27 63 33 69)(28 62 34 68)(29 61 35 67)(30 72 36 66)(37 52 43 58)(38 51 44 57)(39 50 45 56)(40 49 46 55)(41 60 47 54)(42 59 48 53)(73 142 79 136)(74 141 80 135)(75 140 81 134)(76 139 82 133)(77 138 83 144)(78 137 84 143)(85 110 91 116)(86 109 92 115)(87 120 93 114)(88 119 94 113)(89 118 95 112)(90 117 96 111)(97 124 103 130)(98 123 104 129)(99 122 105 128)(100 121 106 127)(101 132 107 126)(102 131 108 125)

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

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

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

54 conjugacy classes

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

54 irreducible representations

 dim 1 1 1 1 1 1 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 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D6 D9 Dic6 D18 D18 S32 S3×Q8 C2×S32 S3×D9 Q8×D9 S3×Dic6 C2×S3×D9 D9×Dic6 kernel D9×Dic6 C9⋊Dic6 Dic3×D9 C9×Dic6 C12×D9 C12.D9 C4×D9 C3×Dic6 C3×D9 Dic9 C36 D18 C3×Dic3 C3×C12 Dic6 D9 Dic3 C12 C12 C32 C6 C4 C3 C3 C2 C1 # reps 1 2 2 1 1 1 1 1 2 1 1 1 2 1 3 4 6 3 1 1 1 3 3 2 3 6

Matrix representation of D9×Dic6 in GL4(𝔽37) generated by

 1 0 0 0 0 1 0 0 0 0 26 31 0 0 6 20
,
 36 0 0 0 0 36 0 0 0 0 31 17 0 0 11 6
,
 32 5 0 0 32 27 0 0 0 0 36 0 0 0 0 36
,
 35 11 0 0 13 2 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(37))| [1,0,0,0,0,1,0,0,0,0,26,6,0,0,31,20],[36,0,0,0,0,36,0,0,0,0,31,11,0,0,17,6],[32,32,0,0,5,27,0,0,0,0,36,0,0,0,0,36],[35,13,0,0,11,2,0,0,0,0,1,0,0,0,0,1] >;

D9×Dic6 in GAP, Magma, Sage, TeX

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

G:=Group("D9xDic6");
// GroupNames label

G:=SmallGroup(432,280);
// by ID

G=gap.SmallGroup(432,280);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,254,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^9=b^2=c^12=1,d^2=c^6,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

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