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## G = C2×C12.D6order 288 = 25·32

### Direct product of C2 and C12.D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12.D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C2×C12.D6
 Lower central C32 — C3×C6 — C2×C12.D6
 Upper central C1 — C22 — C2×D4

Generators and relations for C2×C12.D6
G = < a,b,c,d | a2=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b-1, dcd-1=b6c-1 >

Subgroups: 1508 in 492 conjugacy classes, 165 normal (15 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×2], C4 [×6], C22, C22 [×4], C22 [×8], S3 [×8], C6 [×12], C6 [×16], C2×C4, C2×C4 [×15], D4 [×4], D4 [×8], Q8 [×4], C23 [×2], C23, C32, Dic3 [×24], C12 [×8], D6 [×16], C2×C6 [×20], C2×C6 [×16], C22×C4 [×3], C2×D4, C2×D4 [×2], C2×Q8, C4○D4 [×8], C3⋊S3 [×2], C3×C6, C3×C6 [×2], C3×C6 [×4], Dic6 [×16], C4×S3 [×16], C2×Dic3 [×44], C3⋊D4 [×32], C2×C12 [×4], C3×D4 [×16], C22×S3 [×4], C22×C6 [×8], C2×C4○D4, C3⋊Dic3 [×6], C3×C12 [×2], C2×C3⋊S3 [×2], C2×C3⋊S3 [×2], C62, C62 [×4], C62 [×4], C2×Dic6 [×4], S3×C2×C4 [×4], D42S3 [×32], C22×Dic3 [×8], C2×C3⋊D4 [×8], C6×D4 [×4], C324Q8 [×4], C4×C3⋊S3 [×4], C2×C3⋊Dic3, C2×C3⋊Dic3 [×10], C327D4 [×8], C6×C12, D4×C32 [×4], C22×C3⋊S3, C2×C62 [×2], C2×D42S3 [×4], C2×C324Q8, C2×C4×C3⋊S3, C12.D6 [×8], C22×C3⋊Dic3 [×2], C2×C327D4 [×2], D4×C3×C6, C2×C12.D6
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], D42S3 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×D42S3 [×4], C12.D6 [×2], C23×C3⋊S3, C2×C12.D6

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

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

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

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

60 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 ··· 6L 6M ··· 6AB 12A ··· 12H 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 6 ··· 6 12 ··· 12 size 1 1 1 1 2 2 2 2 18 18 2 2 2 2 2 2 9 9 9 9 18 18 18 18 2 ··· 2 4 ··· 4 4 ··· 4

60 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 4 type + + + + + + + + + + + - image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 D6 C4○D4 D4⋊2S3 kernel C2×C12.D6 C2×C32⋊4Q8 C2×C4×C3⋊S3 C12.D6 C22×C3⋊Dic3 C2×C32⋊7D4 D4×C3×C6 C6×D4 C2×C12 C3×D4 C22×C6 C3×C6 C6 # reps 1 1 1 8 2 2 1 4 4 16 8 4 8

Matrix representation of C2×C12.D6 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
,
 8 0 0 0 0 0 0 5 0 0 0 0 0 0 1 12 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 12 1
,
 0 5 0 0 0 0 8 0 0 0 0 0 0 0 0 12 0 0 0 0 1 12 0 0 0 0 0 0 0 12 0 0 0 0 1 12
,
 0 8 0 0 0 0 8 0 0 0 0 0 0 0 1 12 0 0 0 0 0 12 0 0 0 0 0 0 0 1 0 0 0 0 1 0

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],[8,0,0,0,0,0,0,5,0,0,0,0,0,0,1,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,1],[0,8,0,0,0,0,5,0,0,0,0,0,0,0,0,1,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[0,8,0,0,0,0,8,0,0,0,0,0,0,0,1,0,0,0,0,0,12,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

C2×C12.D6 in GAP, Magma, Sage, TeX

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

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

G:=SmallGroup(288,1008);
// by ID

G=gap.SmallGroup(288,1008);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,2693,9414]);
// 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,c*b*c^-1=b^7,d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations

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