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## G = C2×C9⋊Dic6order 432 = 24·33

### Direct product of C2 and C9⋊Dic6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C18 — C2×C9⋊Dic6
 Chief series C1 — C3 — C32 — C3×C9 — C3×C18 — C9×Dic3 — C9⋊Dic6 — C2×C9⋊Dic6
 Lower central C3×C9 — C3×C18 — C2×C9⋊Dic6
 Upper central C1 — C22

Generators and relations for C2×C9⋊Dic6
G = < a,b,c,d | a2=b9=c12=1, d2=c6, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=c-1 >

Subgroups: 652 in 130 conjugacy classes, 53 normal (25 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C6, C2×C4, Q8, C9, C9, C32, Dic3, Dic3, C12, C2×C6, C2×C6, C2×Q8, C18, C18, C18, C3×C6, C3×C6, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×C9, Dic9, Dic9, C36, C2×C18, C2×C18, C3×Dic3, C3×Dic3, C3⋊Dic3, C62, C2×Dic6, C3×C18, C3×C18, Dic18, C2×Dic9, C2×Dic9, C2×C36, C322Q8, C6×Dic3, C6×Dic3, C2×C3⋊Dic3, C3×Dic9, C9×Dic3, C9⋊Dic3, C6×C18, C2×Dic18, C2×C322Q8, C9⋊Dic6, C6×Dic9, Dic3×C18, C2×C9⋊Dic3, C2×C9⋊Dic6
Quotients: C1, C2, C22, S3, Q8, C23, D6, C2×Q8, D9, Dic6, C22×S3, D18, S32, C2×Dic6, Dic18, C22×D9, C322Q8, C2×S32, S3×D9, C2×Dic18, C2×C322Q8, C9⋊Dic6, C2×S3×D9, C2×C9⋊Dic6

Smallest permutation representation of C2×C9⋊Dic6
On 144 points
Generators in S144
(1 83)(2 84)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 119)(14 120)(15 109)(16 110)(17 111)(18 112)(19 113)(20 114)(21 115)(22 116)(23 117)(24 118)(25 108)(26 97)(27 98)(28 99)(29 100)(30 101)(31 102)(32 103)(33 104)(34 105)(35 106)(36 107)(37 130)(38 131)(39 132)(40 121)(41 122)(42 123)(43 124)(44 125)(45 126)(46 127)(47 128)(48 129)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 85)(59 86)(60 87)(61 142)(62 143)(63 144)(64 133)(65 134)(66 135)(67 136)(68 137)(69 138)(70 139)(71 140)(72 141)
(1 142 110 9 138 118 5 134 114)(2 115 135 6 119 139 10 111 143)(3 144 112 11 140 120 7 136 116)(4 117 137 8 109 141 12 113 133)(13 70 80 17 62 84 21 66 76)(14 77 67 22 73 63 18 81 71)(15 72 82 19 64 74 23 68 78)(16 79 69 24 75 65 20 83 61)(25 59 132 33 55 128 29 51 124)(26 125 52 30 129 56 34 121 60)(27 49 122 35 57 130 31 53 126)(28 127 54 32 131 58 36 123 50)(37 102 92 45 98 88 41 106 96)(38 85 107 42 89 99 46 93 103)(39 104 94 47 100 90 43 108 86)(40 87 97 44 91 101 48 95 105)
(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)
(1 131 7 125)(2 130 8 124)(3 129 9 123)(4 128 10 122)(5 127 11 121)(6 126 12 132)(13 98 19 104)(14 97 20 103)(15 108 21 102)(16 107 22 101)(17 106 23 100)(18 105 24 99)(25 115 31 109)(26 114 32 120)(27 113 33 119)(28 112 34 118)(29 111 35 117)(30 110 36 116)(37 78 43 84)(38 77 44 83)(39 76 45 82)(40 75 46 81)(41 74 47 80)(42 73 48 79)(49 133 55 139)(50 144 56 138)(51 143 57 137)(52 142 58 136)(53 141 59 135)(54 140 60 134)(61 85 67 91)(62 96 68 90)(63 95 69 89)(64 94 70 88)(65 93 71 87)(66 92 72 86)

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

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

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

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 4 4 4 4 4 4 type + + + + + + + - + + + + + - - + + - + - + + - + image C1 C2 C2 C2 C2 S3 S3 Q8 D6 D6 D6 D6 D9 Dic6 Dic6 D18 D18 Dic18 S32 C32⋊2Q8 C2×S32 S3×D9 C9⋊Dic6 C2×S3×D9 kernel C2×C9⋊Dic6 C9⋊Dic6 C6×Dic9 Dic3×C18 C2×C9⋊Dic3 C2×Dic9 C6×Dic3 C3×C18 Dic9 C2×C18 C3×Dic3 C62 C2×Dic3 C18 C3×C6 Dic3 C2×C6 C6 C2×C6 C6 C6 C22 C2 C2 # reps 1 4 1 1 1 1 1 2 2 1 2 1 3 4 4 6 3 12 1 2 1 3 6 3

Matrix representation of C2×C9⋊Dic6 in GL6(𝔽37)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36 0 0 0 0 0 0 36
,
 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 31 27 0 0 0 0 3 11
,
 20 2 0 0 0 0 3 17 0 0 0 0 0 0 0 36 0 0 0 0 1 36 0 0 0 0 0 0 36 0 0 0 0 0 35 1
,
 29 15 0 0 0 0 8 8 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

G:=sub<GL(6,GF(37))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36,0,0,0,0,0,0,36],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,31,3,0,0,0,0,27,11],[20,3,0,0,0,0,2,17,0,0,0,0,0,0,0,1,0,0,0,0,36,36,0,0,0,0,0,0,36,35,0,0,0,0,0,1],[29,8,0,0,0,0,15,8,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;

C2×C9⋊Dic6 in GAP, Magma, Sage, TeX

C_2\times C_9\rtimes {\rm Dic}_6
% in TeX

G:=Group("C2xC9:Dic6");
// GroupNames label

G:=SmallGroup(432,303);
// by ID

G=gap.SmallGroup(432,303);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-3,56,141,64,3091,662,4037,7069]);
// Polycyclic

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

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