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## G = D6.Dic9order 432 = 24·33

### The non-split extension by D6 of Dic9 acting via Dic9/C18=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — D6.Dic9
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C3×C36 — S3×C36 — D6.Dic9
 Lower central C3×C9 — C3×C18 — D6.Dic9
 Upper central C1 — C4

Generators and relations for D6.Dic9
G = < a,b,c,d | a6=b2=1, c18=a3, d2=c9, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=c17 >

Subgroups: 244 in 68 conjugacy classes, 31 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C2×C4, C9, C9, C32, Dic3, C12, C12, D6, C2×C6, M4(2), C18, C18, C3×S3, C3×C6, C3⋊C8, C24, C4×S3, C2×C12, C3×C9, C36, C36, C2×C18, C3×Dic3, C3×C12, S3×C6, C8⋊S3, C4.Dic3, S3×C9, C3×C18, C9⋊C8, C9⋊C8, C2×C36, C3×C3⋊C8, C324C8, S3×C12, C9×Dic3, C3×C36, S3×C18, C4.Dic9, D6.Dic3, C3×C9⋊C8, C36.S3, S3×C36, D6.Dic9
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, M4(2), D9, C4×S3, C2×Dic3, Dic9, D18, S32, C8⋊S3, C4.Dic3, C2×Dic9, S3×Dic3, S3×D9, C4.Dic9, D6.Dic3, S3×Dic9, D6.Dic9

Smallest permutation representation of D6.Dic9
On 144 points
Generators in S144
(1 7 13 19 25 31)(2 8 14 20 26 32)(3 9 15 21 27 33)(4 10 16 22 28 34)(5 11 17 23 29 35)(6 12 18 24 30 36)(37 67 61 55 49 43)(38 68 62 56 50 44)(39 69 63 57 51 45)(40 70 64 58 52 46)(41 71 65 59 53 47)(42 72 66 60 54 48)(73 103 97 91 85 79)(74 104 98 92 86 80)(75 105 99 93 87 81)(76 106 100 94 88 82)(77 107 101 95 89 83)(78 108 102 96 90 84)(109 115 121 127 133 139)(110 116 122 128 134 140)(111 117 123 129 135 141)(112 118 124 130 136 142)(113 119 125 131 137 143)(114 120 126 132 138 144)
(1 106)(2 107)(3 108)(4 73)(5 74)(6 75)(7 76)(8 77)(9 78)(10 79)(11 80)(12 81)(13 82)(14 83)(15 84)(16 85)(17 86)(18 87)(19 88)(20 89)(21 90)(22 91)(23 92)(24 93)(25 94)(26 95)(27 96)(28 97)(29 98)(30 99)(31 100)(32 101)(33 102)(34 103)(35 104)(36 105)(37 131)(38 132)(39 133)(40 134)(41 135)(42 136)(43 137)(44 138)(45 139)(46 140)(47 141)(48 142)(49 143)(50 144)(51 109)(52 110)(53 111)(54 112)(55 113)(56 114)(57 115)(58 116)(59 117)(60 118)(61 119)(62 120)(63 121)(64 122)(65 123)(66 124)(67 125)(68 126)(69 127)(70 128)(71 129)(72 130)
(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 46 10 55 19 64 28 37)(2 63 11 72 20 45 29 54)(3 44 12 53 21 62 30 71)(4 61 13 70 22 43 31 52)(5 42 14 51 23 60 32 69)(6 59 15 68 24 41 33 50)(7 40 16 49 25 58 34 67)(8 57 17 66 26 39 35 48)(9 38 18 47 27 56 36 65)(73 137 82 110 91 119 100 128)(74 118 83 127 92 136 101 109)(75 135 84 144 93 117 102 126)(76 116 85 125 94 134 103 143)(77 133 86 142 95 115 104 124)(78 114 87 123 96 132 105 141)(79 131 88 140 97 113 106 122)(80 112 89 121 98 130 107 139)(81 129 90 138 99 111 108 120)

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

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

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

66 conjugacy classes

 class 1 2A 2B 3A 3B 3C 4A 4B 4C 6A 6B 6C 6D 6E 8A 8B 8C 8D 9A 9B 9C 9D 9E 9F 12A 12B 12C 12D 12E 12F 12G 12H 18A 18B 18C 18D 18E 18F 18G ··· 18L 24A 24B 24C 24D 36A ··· 36F 36G ··· 36L 36M ··· 36R order 1 2 2 3 3 3 4 4 4 6 6 6 6 6 8 8 8 8 9 9 9 9 9 9 12 12 12 12 12 12 12 12 18 18 18 18 18 18 18 ··· 18 24 24 24 24 36 ··· 36 36 ··· 36 36 ··· 36 size 1 1 6 2 2 4 1 1 6 2 2 4 6 6 18 18 54 54 2 2 2 4 4 4 2 2 2 2 4 4 6 6 2 2 2 4 4 4 6 ··· 6 18 18 18 18 2 ··· 2 4 ··· 4 6 ··· 6

66 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4 4 4 4 4 4 type + + + + + + + - + - + - + - + - + - image C1 C2 C2 C2 C4 C4 S3 S3 D6 Dic3 D6 Dic3 M4(2) D9 C4×S3 Dic9 D18 Dic9 C8⋊S3 C4.Dic3 C4.Dic9 S32 S3×Dic3 S3×D9 D6.Dic3 S3×Dic9 D6.Dic9 kernel D6.Dic9 C3×C9⋊C8 C36.S3 S3×C36 C9×Dic3 S3×C18 C9⋊C8 S3×C12 C36 C3×Dic3 C3×C12 S3×C6 C3×C9 C4×S3 C18 Dic3 C12 D6 C9 C32 C3 C12 C6 C4 C3 C2 C1 # reps 1 1 1 1 2 2 1 1 1 1 1 1 2 3 2 3 3 3 4 4 12 1 1 3 2 3 6

Matrix representation of D6.Dic9 in GL4(𝔽73) generated by

 1 72 0 0 1 0 0 0 0 0 1 0 0 0 0 1
,
 50 5 0 0 55 23 0 0 0 0 1 0 0 0 0 1
,
 27 0 0 0 0 27 0 0 0 0 28 26 0 0 22 70
,
 65 16 0 0 57 8 0 0 0 0 46 0 0 0 35 27
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[50,55,0,0,5,23,0,0,0,0,1,0,0,0,0,1],[27,0,0,0,0,27,0,0,0,0,28,22,0,0,26,70],[65,57,0,0,16,8,0,0,0,0,46,35,0,0,0,27] >;

D6.Dic9 in GAP, Magma, Sage, TeX

D_6.{\rm Dic}_9
% in TeX

G:=Group("D6.Dic9");
// GroupNames label

G:=SmallGroup(432,67);
// by ID

G=gap.SmallGroup(432,67);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,141,36,58,3091,662,4037,7069]);
// Polycyclic

G:=Group<a,b,c,d|a^6=b^2=1,c^18=a^3,d^2=c^9,b*a*b=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^17>;
// generators/relations

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