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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, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C3×Q8, C22×S3, C2×C4○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, S3×C2×C4, C2×D12, Q83S3, C6×Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, Q8×C32, C22×C3⋊S3, C2×Q83S3, C2×C4×C3⋊S3, C2×C12⋊S3, C12.26D6, Q8×C3×C6, C2×C12.26D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, Q83S3, S3×C23, C22×C3⋊S3, C2×Q83S3, C12.26D6, C23×C3⋊S3, C2×C12.26D6

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

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

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

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

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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