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G = C2×C12.D6order 288 = 25·32

Direct product of C2 and C12.D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12.D6, C62.278C23, (C6×D4)⋊6S3, (C3×D4)⋊17D6, C65(D42S3), (C2×C12).169D6, C6.59(S3×C23), (C3×C6).58C24, (C22×C6).101D6, C12.110(C22×S3), (C3×C12).129C23, (C6×C12).168C22, (D4×C32)⋊24C22, C327D412C22, C3⋊Dic3.47C23, (C2×C62).84C22, C324Q823C22, D45(C2×C3⋊S3), (D4×C3×C6)⋊13C2, (C2×D4)⋊8(C3⋊S3), C36(C2×D42S3), C3216(C2×C4○D4), C2.7(C23×C3⋊S3), (C3×C6)⋊10(C4○D4), (C4×C3⋊S3)⋊15C22, C23.24(C2×C3⋊S3), C4.20(C22×C3⋊S3), (C2×C3⋊S3).51C23, (C2×C327D4)⋊19C2, (C2×C324Q8)⋊21C2, C22.1(C22×C3⋊S3), (C2×C6).287(C22×S3), (C22×C3⋊Dic3)⋊15C2, (C2×C3⋊Dic3)⋊27C22, (C22×C3⋊S3).107C22, (C2×C4×C3⋊S3)⋊8C2, (C2×C4).60(C2×C3⋊S3), SmallGroup(288,1008)

Series: Derived Chief Lower central Upper central

C1C3×C6 — C2×C12.D6
C1C3C32C3×C6C2×C3⋊S3C22×C3⋊S3C2×C4×C3⋊S3 — C2×C12.D6
C32C3×C6 — C2×C12.D6
C1C22C2×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 2A2B2C2D2E2F2G2H2I3A3B3C3D4A4B4C4D4E4F4G4H4I4J6A···6L6M···6AB12A···12H
order1222222222333344444444446···66···612···12
size1111222218182222229999181818182···24···44···4

60 irreducible representations

dim1111111222224
type+++++++++++-
imageC1C2C2C2C2C2C2S3D6D6D6C4○D4D42S3
kernelC2×C12.D6C2×C324Q8C2×C4×C3⋊S3C12.D6C22×C3⋊Dic3C2×C327D4D4×C3×C6C6×D4C2×C12C3×D4C22×C6C3×C6C6
# reps11182214416848

Matrix representation of C2×C12.D6 in GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
800000
050000
0011200
001000
000001
0000121
,
050000
800000
0001200
0011200
0000012
0000112
,
080000
800000
0011200
0001200
000001
000010

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