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## G = C3×C12.26D6order 432 = 24·33

### Direct product of C3 and C12.26D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C3×C12.26D6
 Chief series C1 — C3 — C32 — C3×C6 — C32×C6 — C6×C3⋊S3 — C12×C3⋊S3 — C3×C12.26D6
 Lower central C32 — C3×C6 — C3×C12.26D6
 Upper central C1 — C6 — C3×Q8

Generators and relations for C3×C12.26D6
G = < a,b,c,d | a3=b12=1, c6=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c5 >

Subgroups: 916 in 288 conjugacy classes, 94 normal (16 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C32, C32, C32, Dic3, C12, C12, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, Q83S3, C3×C4○D4, C3×C3⋊S3, C32×C6, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, Q8×C32, Q8×C32, Q8×C32, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×Q83S3, C12.26D6, C12×C3⋊S3, C3×C12⋊S3, Q8×C33, C3×C12.26D6
Quotients: C1, C2, C3, C22, S3, C6, C23, D6, C2×C6, C4○D4, C3×S3, C3⋊S3, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, Q83S3, C3×C4○D4, C3×C3⋊S3, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C3×Q83S3, C12.26D6, C2×C6×C3⋊S3, C3×C12.26D6

Smallest permutation representation of C3×C12.26D6
On 144 points
Generators in S144
(1 91 38)(2 92 39)(3 93 40)(4 94 41)(5 95 42)(6 96 43)(7 85 44)(8 86 45)(9 87 46)(10 88 47)(11 89 48)(12 90 37)(13 117 105)(14 118 106)(15 119 107)(16 120 108)(17 109 97)(18 110 98)(19 111 99)(20 112 100)(21 113 101)(22 114 102)(23 115 103)(24 116 104)(25 131 61)(26 132 62)(27 121 63)(28 122 64)(29 123 65)(30 124 66)(31 125 67)(32 126 68)(33 127 69)(34 128 70)(35 129 71)(36 130 72)(49 134 73)(50 135 74)(51 136 75)(52 137 76)(53 138 77)(54 139 78)(55 140 79)(56 141 80)(57 142 81)(58 143 82)(59 144 83)(60 133 84)
(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)
(1 108 48 118 87 24 7 102 42 112 93 18)(2 103 37 113 88 19 8 97 43 119 94 13)(3 98 38 120 89 14 9 104 44 114 95 20)(4 105 39 115 90 21 10 99 45 109 96 15)(5 100 40 110 91 16 11 106 46 116 85 22)(6 107 41 117 92 23 12 101 47 111 86 17)(25 59 129 142 69 79 31 53 123 136 63 73)(26 54 130 137 70 74 32 60 124 143 64 80)(27 49 131 144 71 81 33 55 125 138 65 75)(28 56 132 139 72 76 34 50 126 133 66 82)(29 51 121 134 61 83 35 57 127 140 67 77)(30 58 122 141 62 78 36 52 128 135 68 84)
(1 77 7 83)(2 82 8 76)(3 75 9 81)(4 80 10 74)(5 73 11 79)(6 78 12 84)(13 34 19 28)(14 27 20 33)(15 32 21 26)(16 25 22 31)(17 30 23 36)(18 35 24 29)(37 133 43 139)(38 138 44 144)(39 143 45 137)(40 136 46 142)(41 141 47 135)(42 134 48 140)(49 89 55 95)(50 94 56 88)(51 87 57 93)(52 92 58 86)(53 85 59 91)(54 90 60 96)(61 102 67 108)(62 107 68 101)(63 100 69 106)(64 105 70 99)(65 98 71 104)(66 103 72 97)(109 124 115 130)(110 129 116 123)(111 122 117 128)(112 127 118 121)(113 132 119 126)(114 125 120 131)

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

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

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

90 conjugacy classes

 class 1 2A 2B 2C 2D 3A 3B 3C ··· 3N 4A 4B 4C 4D 4E 6A 6B 6C ··· 6N 6O ··· 6T 12A ··· 12F 12G ··· 12AP 12AQ 12AR 12AS 12AT order 1 2 2 2 2 3 3 3 ··· 3 4 4 4 4 4 6 6 6 ··· 6 6 ··· 6 12 ··· 12 12 ··· 12 12 12 12 12 size 1 1 18 18 18 1 1 2 ··· 2 2 2 2 9 9 1 1 2 ··· 2 18 ··· 18 2 ··· 2 4 ··· 4 9 9 9 9

90 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C2 C3 C6 C6 C6 S3 D6 C4○D4 C3×S3 S3×C6 C3×C4○D4 Q8⋊3S3 C3×Q8⋊3S3 kernel C3×C12.26D6 C12×C3⋊S3 C3×C12⋊S3 Q8×C33 C12.26D6 C4×C3⋊S3 C12⋊S3 Q8×C32 Q8×C32 C3×C12 C33 C3×Q8 C12 C32 C32 C3 # reps 1 3 3 1 2 6 6 2 4 12 2 8 24 4 4 8

Matrix representation of C3×C12.26D6 in GL6(𝔽13)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 9 0 0 0 0 0 0 9 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 3 0 0 0 0 0 0 9 0 0 0 0 0 0 10 0 0 0 0 0 8 4 0 0 0 0 0 0 4 3 0 0 0 0 3 9
,
 4 0 0 0 0 0 0 10 0 0 0 0 0 0 3 0 0 0 0 0 5 9 0 0 0 0 0 0 7 1 0 0 0 0 2 6
,
 0 12 0 0 0 0 12 0 0 0 0 0 0 0 6 2 0 0 0 0 2 7 0 0 0 0 0 0 8 0 0 0 0 0 0 8

G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,10,8,0,0,0,0,0,4,0,0,0,0,0,0,4,3,0,0,0,0,3,9],[4,0,0,0,0,0,0,10,0,0,0,0,0,0,3,5,0,0,0,0,0,9,0,0,0,0,0,0,7,2,0,0,0,0,1,6],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,6,2,0,0,0,0,2,7,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;

C3×C12.26D6 in GAP, Magma, Sage, TeX

C_3\times C_{12}._{26}D_6
% in TeX

G:=Group("C3xC12.26D6");
// GroupNames label

G:=SmallGroup(432,717);
// by ID

G=gap.SmallGroup(432,717);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,176,590,303,142,4037,14118]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^12=1,c^6=d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^7,d*b*d^-1=b^5,d*c*d^-1=c^5>;
// generators/relations

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