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G = C2×C12.26D6  order 288 = 25·32

Direct product of C2 and C12.26D6

direct product, metabelian, supersoluble, monomial

Aliases: C2×C12.26D6, C62.281C23, (C3×Q8)⋊19D6, (C6×Q8)⋊10S3, C6⋊3(Q8⋊3S3), (C2×C12).171D6, C6.62(S3×C23), (C3×C6).61C24, C12⋊S3⋊27C22, C12.113(C22×S3), (C3×C12).132C23, (C6×C12).170C22, C3⋊Dic3.54C23, (Q8×C32)⋊22C22, Q8⋊6(C2×C3⋊S3), (Q8×C3×C6)⋊13C2, (C2×Q8)⋊8(C3⋊S3), C3⋊4(C2×Q8⋊3S3), C32⋊18(C2×C4○D4), (C3×C6)⋊12(C4○D4), (C4×C3⋊S3)⋊16C22, (C2×C12⋊S3)⋊21C2, C4.23(C22×C3⋊S3), C2.10(C23×C3⋊S3), (C2×C3⋊S3).53C23, (C2×C6).289(C22×S3), C22.32(C22×C3⋊S3), (C22×C3⋊S3).108C22, (C2×C3⋊Dic3).188C22, (C2×C4×C3⋊S3)⋊9C2, (C2×C4).62(C2×C3⋊S3), SmallGroup(288,1011)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C3×C6 — C2×C12.26D6
Chief series
C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C2×C12.26D6
Lower central
C32 — C3×C6 — C2×C12.26D6

Subgroups: 1764 in 492 conjugacy classes, 165 normal (10 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×6], C4 [×2], C22, C22 [×12], S3 [×24], C6 [×12], C2×C4 [×3], C2×C4 [×13], D4 [×12], Q8 [×4], C23 [×3], C32, Dic3 [×8], C12 [×24], D6 [×48], C2×C6 [×4], C22×C4 [×3], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊S3 [×6], C3×C6, C3×C6 [×2], C4×S3 [×48], D12 [×48], C2×Dic3 [×4], C2×C12 [×12], C3×Q8 [×16], C22×S3 [×12], C2×C4○D4, C3⋊Dic3 [×2], C3×C12 [×6], C2×C3⋊S3 [×6], C2×C3⋊S3 [×6], C62, S3×C2×C4 [×12], C2×D12 [×12], Q8⋊3S3 [×32], C6×Q8 [×4], C4×C3⋊S3 [×12], C12⋊S3 [×12], C2×C3⋊Dic3, C6×C12 [×3], Q8×C32 [×4], C22×C3⋊S3 [×3], C2×Q8⋊3S3 [×4], C2×C4×C3⋊S3 [×3], C2×C12⋊S3 [×3], C12.26D6 [×8], Q8×C3×C6, C2×C12.26D6

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], Q8⋊3S3 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×Q8⋊3S3 [×4], C12.26D6 [×2], C23×C3⋊S3, C2×C12.26D6

Generators and relations
 G = < a,b,c,d | a2=b12=1, c6=d2=b6, ab=ba, ac=ca, ad=da, cbc-1=b7, dbd-1=b5, dcd-1=c5 >

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

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

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

Matrix representation ►G ⊆ GL6(𝔽13)

1200000
0120000
0012000
0001200
0000120
0000012
,
0120000
110000
000100
00121200
000013
0000812
,
12120000
100000
001000
000100
000052
000008
,
010000
100000
0012000
001100
000080
000008

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],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,8,0,0,0,0,3,12],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,5,0,0,0,0,0,2,8],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,8,0,0,0,0,0,0,8] >;
 

60 conjugacy classes

class 1 2A2B2C2D···2I3A3B3C3D4A···4F4G4H4I4J6A···6L12A···12X
order12222···233334···444446···612···12
size111118···1822222···299992···24···4

60 irreducible representations

dim1111122224
type+++++++++
imageC1C2C2C2C2S3D6D6C4○D4Q8⋊3S3
kernelC2×C12.26D6C2×C4×C3⋊S3C2×C12⋊S3C12.26D6Q8×C3×C6C6×Q8C2×C12C3×Q8C3×C6C6
# reps133814121648

In GAP, Magma, Sage, TeX 

C_2\times C_{12}._{26}D_6
 
% in TeX
 
G:=Group("C2xC12.26D6");
 
// GroupNames label
 
G:=SmallGroup(288,1011);
 
// by ID
 
G=gap.SmallGroup(288,1011);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,185,80,2693,9414]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^12=1,c^6=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^5,d*c*d^-1=c^5>;
 
// generators/relations
 

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