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## G = C2×C12.59D6order 288 = 25·32

### Direct product of C2 and C12.59D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12.59D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C2×C12.59D6
 Lower central C32 — C3×C6 — C2×C12.59D6
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C12.59D6
G = < a,b,c,d | a2=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 1636 in 492 conjugacy classes, 165 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, C2×C4, D4, Q8, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C22×S3, C22×C6, C2×C4○D4, C3⋊Dic3, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C22×C12, C324Q8, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C327D4, C6×C12, C6×C12, C22×C3⋊S3, C2×C62, C2×C4○D12, C2×C324Q8, C2×C4×C3⋊S3, C2×C12⋊S3, C12.59D6, C2×C327D4, C2×C6×C12, C2×C12.59D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, C4○D12, S3×C23, C22×C3⋊S3, C2×C4○D12, C12.59D6, C23×C3⋊S3, C2×C12.59D6

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

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

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

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

84 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 3A 3B 3C 3D 4A 4B 4C 4D 4E 4F 4G 4H 4I 4J 6A ··· 6AB 12A ··· 12AF order 1 2 2 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 4 4 4 4 4 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 18 18 18 18 2 2 2 2 1 1 1 1 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 kernel C2×C12.59D6 C2×C32⋊4Q8 C2×C4×C3⋊S3 C2×C12⋊S3 C12.59D6 C2×C32⋊7D4 C2×C6×C12 C22×C12 C2×C12 C22×C6 C3×C6 C6 # reps 1 1 2 1 8 2 1 4 24 4 4 32

Matrix representation of C2×C12.59D6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 5 8 0 0 5 0
,
 1 1 0 0 12 0 0 0 0 0 4 11 0 0 2 2
,
 12 12 0 0 0 1 0 0 0 0 9 2 0 0 11 4
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,5,0,0,8,0],[1,12,0,0,1,0,0,0,0,0,4,2,0,0,11,2],[12,0,0,0,12,1,0,0,0,0,9,11,0,0,2,4] >;

C2×C12.59D6 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{59}D_6
% in TeX

G:=Group("C2xC12.59D6");
// GroupNames label

G:=SmallGroup(288,1006);
// by ID

G=gap.SmallGroup(288,1006);
# by ID

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

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

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