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## G = C2×C6.11D12order 288 = 25·32

### Direct product of C2 and C6.11D12

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C6.11D12
 Chief series C1 — C3 — C32 — C3×C6 — C62 — C22×C3⋊S3 — C23×C3⋊S3 — C2×C6.11D12
 Lower central C32 — C3×C6 — C2×C6.11D12
 Upper central C1 — C23 — C22×C4

Generators and relations for C2×C6.11D12
G = < a,b,c,d | a2=b6=c12=1, d2=b3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=b3c-1 >

Subgroups: 1636 in 396 conjugacy classes, 133 normal (17 characteristic)
C1, C2 [×3], C2 [×4], C2 [×4], C3 [×4], C4 [×4], C22, C22 [×6], C22 [×16], S3 [×16], C6 [×28], C2×C4 [×2], C2×C4 [×6], C23, C23 [×10], C32, Dic3 [×8], C12 [×8], D6 [×64], C2×C6 [×28], C22⋊C4 [×4], C22×C4, C22×C4, C24, C3⋊S3 [×4], C3×C6 [×3], C3×C6 [×4], C2×Dic3 [×16], C2×C12 [×8], C2×C12 [×8], C22×S3 [×40], C22×C6 [×4], C2×C22⋊C4, C3⋊Dic3 [×2], C3×C12 [×2], C2×C3⋊S3 [×4], C2×C3⋊S3 [×12], C62, C62 [×6], D6⋊C4 [×16], C22×Dic3 [×4], C22×C12 [×4], S3×C23 [×4], C2×C3⋊Dic3 [×2], C2×C3⋊Dic3 [×2], C6×C12 [×2], C6×C12 [×2], C22×C3⋊S3 [×6], C22×C3⋊S3 [×4], C2×C62, C2×D6⋊C4 [×4], C6.11D12 [×4], C22×C3⋊Dic3, C2×C6×C12, C23×C3⋊S3, C2×C6.11D12
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], S3 [×4], C2×C4 [×6], D4 [×4], C23, D6 [×12], C22⋊C4 [×4], C22×C4, C2×D4 [×2], C3⋊S3, C4×S3 [×8], D12 [×8], C3⋊D4 [×8], C22×S3 [×4], C2×C22⋊C4, C2×C3⋊S3 [×3], D6⋊C4 [×16], S3×C2×C4 [×4], C2×D12 [×4], C2×C3⋊D4 [×4], C4×C3⋊S3 [×2], C12⋊S3 [×2], C327D4 [×2], C22×C3⋊S3, C2×D6⋊C4 [×4], C6.11D12 [×4], C2×C4×C3⋊S3, C2×C12⋊S3, C2×C327D4, C2×C6.11D12

Smallest permutation representation of C2×C6.11D12
On 144 points
Generators in S144
(1 102)(2 103)(3 104)(4 105)(5 106)(6 107)(7 108)(8 97)(9 98)(10 99)(11 100)(12 101)(13 114)(14 115)(15 116)(16 117)(17 118)(18 119)(19 120)(20 109)(21 110)(22 111)(23 112)(24 113)(25 72)(26 61)(27 62)(28 63)(29 64)(30 65)(31 66)(32 67)(33 68)(34 69)(35 70)(36 71)(37 128)(38 129)(39 130)(40 131)(41 132)(42 121)(43 122)(44 123)(45 124)(46 125)(47 126)(48 127)(49 88)(50 89)(51 90)(52 91)(53 92)(54 93)(55 94)(56 95)(57 96)(58 85)(59 86)(60 87)(73 143)(74 144)(75 133)(76 134)(77 135)(78 136)(79 137)(80 138)(81 139)(82 140)(83 141)(84 142)
(1 76 89 17 126 31)(2 77 90 18 127 32)(3 78 91 19 128 33)(4 79 92 20 129 34)(5 80 93 21 130 35)(6 81 94 22 131 36)(7 82 95 23 132 25)(8 83 96 24 121 26)(9 84 85 13 122 27)(10 73 86 14 123 28)(11 74 87 15 124 29)(12 75 88 16 125 30)(37 68 104 136 52 120)(38 69 105 137 53 109)(39 70 106 138 54 110)(40 71 107 139 55 111)(41 72 108 140 56 112)(42 61 97 141 57 113)(43 62 98 142 58 114)(44 63 99 143 59 115)(45 64 100 144 60 116)(46 65 101 133 49 117)(47 66 102 134 50 118)(48 67 103 135 51 119)
(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 16 17 12)(2 11 18 15)(3 14 19 10)(4 9 20 13)(5 24 21 8)(6 7 22 23)(25 131 95 81)(26 80 96 130)(27 129 85 79)(28 78 86 128)(29 127 87 77)(30 76 88 126)(31 125 89 75)(32 74 90 124)(33 123 91 73)(34 84 92 122)(35 121 93 83)(36 82 94 132)(37 63 136 59)(38 58 137 62)(39 61 138 57)(40 56 139 72)(41 71 140 55)(42 54 141 70)(43 69 142 53)(44 52 143 68)(45 67 144 51)(46 50 133 66)(47 65 134 49)(48 60 135 64)(97 106 113 110)(98 109 114 105)(99 104 115 120)(100 119 116 103)(101 102 117 118)(107 108 111 112)

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

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

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

84 conjugacy classes

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

84 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C4 S3 D4 D6 D6 C4×S3 D12 C3⋊D4 kernel C2×C6.11D12 C6.11D12 C22×C3⋊Dic3 C2×C6×C12 C23×C3⋊S3 C22×C3⋊S3 C22×C12 C62 C2×C12 C22×C6 C2×C6 C2×C6 C2×C6 # reps 1 4 1 1 1 8 4 4 8 4 16 16 16

Matrix representation of C2×C6.11D12 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
,
 12 12 0 0 0 0 1 0 0 0 0 0 0 0 12 0 0 0 0 0 0 12 0 0 0 0 0 0 0 12 0 0 0 0 1 1
,
 3 6 0 0 0 0 7 10 0 0 0 0 0 0 9 10 0 0 0 0 5 4 0 0 0 0 0 0 8 8 0 0 0 0 5 0
,
 3 6 0 0 0 0 3 10 0 0 0 0 0 0 9 10 0 0 0 0 10 4 0 0 0 0 0 0 5 0 0 0 0 0 8 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],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,1],[3,7,0,0,0,0,6,10,0,0,0,0,0,0,9,5,0,0,0,0,10,4,0,0,0,0,0,0,8,5,0,0,0,0,8,0],[3,3,0,0,0,0,6,10,0,0,0,0,0,0,9,10,0,0,0,0,10,4,0,0,0,0,0,0,5,8,0,0,0,0,0,8] >;

C2×C6.11D12 in GAP, Magma, Sage, TeX

C_2\times C_6._{11}D_{12}
% in TeX

G:=Group("C2xC6.11D12");
// GroupNames label

G:=SmallGroup(288,784);
// by ID

G=gap.SmallGroup(288,784);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,422,58,2693,9414]);
// Polycyclic

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

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