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## G = C2×C12.26D6order 288 = 25·32

### Direct product of C2 and C12.26D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12.26D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C2×C12.26D6
 Lower central C32 — C3×C6 — C2×C12.26D6
 Upper central C1 — C22 — C2×Q8

Generators and relations for C2×C12.26D6
G = < a,b,c,d | a2=b12=1, c6=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c5 >

Subgroups: 1764 in 492 conjugacy classes, 165 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×6], C4 [×2], C22, C22 [×12], S3 [×24], C6 [×12], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×8], C12 [×24], D6 [×48], C2×C6 [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊S3 [×6], C3×C6, C3×C6 [×2], C4×S3 [×48], D12 [×48], C2×Dic3 [×4], C2×C12 [×12], C3×Q8 [×16], C22×S3 [×12], C2×C4○D4, C3⋊Dic3 [×2], C3×C12 [×6], C2×C3⋊S3 [×6], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×12], C2×D12 [×12], Q83S3 [×32], C6×Q8 [×4], C4×C3⋊S3 [×12], C12⋊S3 [×12], C2×C3⋊Dic3, C6×C12 [×3], Q8×C32 [×4], C22×C3⋊S3 [×3], C2×Q83S3 [×4], C2×C4×C3⋊S3 [×3], C2×C12⋊S3 [×3], C12.26D6 [×8], Q8×C3×C6, C2×C12.26D6
Quotients: C1, C2 [×15], C22 [×35], S3 [×4], C23 [×15], D6 [×28], C4○D4 [×2], C24, C3⋊S3, C22×S3 [×28], C2×C4○D4, C2×C3⋊S3 [×7], Q83S3 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×Q83S3 [×4], C12.26D6 [×2], C23×C3⋊S3, C2×C12.26D6

Smallest permutation representation of C2×C12.26D6
On 144 points
Generators in S144
(1 106)(2 107)(3 108)(4 97)(5 98)(6 99)(7 100)(8 101)(9 102)(10 103)(11 104)(12 105)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 117)(26 118)(27 119)(28 120)(29 109)(30 110)(31 111)(32 112)(33 113)(34 114)(35 115)(36 116)(37 84)(38 73)(39 74)(40 75)(41 76)(42 77)(43 78)(44 79)(45 80)(46 81)(47 82)(48 83)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(55 127)(56 128)(57 129)(58 130)(59 131)(60 132)(61 133)(62 134)(63 135)(64 136)(65 137)(66 138)(67 139)(68 140)(69 141)(70 142)(71 143)(72 144)
(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)
(1 134 49 109 24 44 7 140 55 115 18 38)(2 141 50 116 13 39 8 135 56 110 19 45)(3 136 51 111 14 46 9 142 57 117 20 40)(4 143 52 118 15 41 10 137 58 112 21 47)(5 138 53 113 16 48 11 144 59 119 22 42)(6 133 54 120 17 43 12 139 60 114 23 37)(25 92 75 108 64 123 31 86 81 102 70 129)(26 87 76 103 65 130 32 93 82 97 71 124)(27 94 77 98 66 125 33 88 83 104 72 131)(28 89 78 105 67 132 34 95 84 99 61 126)(29 96 79 100 68 127 35 90 73 106 62 121)(30 91 80 107 69 122 36 85 74 101 63 128)
(1 76 7 82)(2 81 8 75)(3 74 9 80)(4 79 10 73)(5 84 11 78)(6 77 12 83)(13 64 19 70)(14 69 20 63)(15 62 21 68)(16 67 22 61)(17 72 23 66)(18 65 24 71)(25 50 31 56)(26 55 32 49)(27 60 33 54)(28 53 34 59)(29 58 35 52)(30 51 36 57)(37 104 43 98)(38 97 44 103)(39 102 45 108)(40 107 46 101)(41 100 47 106)(42 105 48 99)(85 136 91 142)(86 141 92 135)(87 134 93 140)(88 139 94 133)(89 144 95 138)(90 137 96 143)(109 130 115 124)(110 123 116 129)(111 128 117 122)(112 121 118 127)(113 126 119 132)(114 131 120 125)

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

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

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

60 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 3A 3B 3C 3D 4A ··· 4F 4G 4H 4I 4J 6A ··· 6L 12A ··· 12X order 1 2 2 2 2 ··· 2 3 3 3 3 4 ··· 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 18 ··· 18 2 2 2 2 2 ··· 2 9 9 9 9 2 ··· 2 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 4 type + + + + + + + + + image C1 C2 C2 C2 C2 S3 D6 D6 C4○D4 Q8⋊3S3 kernel C2×C12.26D6 C2×C4×C3⋊S3 C2×C12⋊S3 C12.26D6 Q8×C3×C6 C6×Q8 C2×C12 C3×Q8 C3×C6 C6 # reps 1 3 3 8 1 4 12 16 4 8

Matrix representation of C2×C12.26D6 in GL6(𝔽13)

 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 12
,
 0 12 0 0 0 0 1 1 0 0 0 0 0 0 0 1 0 0 0 0 12 12 0 0 0 0 0 0 1 3 0 0 0 0 8 12
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 5 2 0 0 0 0 0 8
,
 0 1 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 1 1 0 0 0 0 0 0 8 0 0 0 0 0 0 8

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

C2×C12.26D6 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{26}D_6
% in TeX

G:=Group("C2xC12.26D6");
// GroupNames label

G:=SmallGroup(288,1011);
// by ID

G=gap.SmallGroup(288,1011);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,80,2693,9414]);
// Polycyclic

G:=Group<a,b,c,d|a^2=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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