Copied to
clipboard

## 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, C2, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, D4, Q8, C23, C23, C32, Dic3, C12, D6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C2×Q8, C4○D4, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C3×D4, 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, D42S3, C22×Dic3, C2×C3⋊D4, C6×D4, C324Q8, C4×C3⋊S3, C2×C3⋊Dic3, C2×C3⋊Dic3, C327D4, C6×C12, D4×C32, C22×C3⋊S3, C2×C62, C2×D42S3, C2×C324Q8, C2×C4×C3⋊S3, C12.D6, C22×C3⋊Dic3, C2×C327D4, D4×C3×C6, C2×C12.D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C3⋊S3, C22×S3, C2×C4○D4, C2×C3⋊S3, D42S3, S3×C23, C22×C3⋊S3, C2×D42S3, C12.D6, C23×C3⋊S3, C2×C12.D6

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

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

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

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

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

׿
×
𝔽