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## G = C2×C12.59D6order 288 = 25·32

### Direct product of C2 and C12.59D6

Series: Derived Chief Lower central Upper central

 Derived series C1 — C3×C6 — C2×C12.59D6
 Chief series C1 — C3 — C32 — C3×C6 — C2×C3⋊S3 — C22×C3⋊S3 — C2×C4×C3⋊S3 — C2×C12.59D6
 Lower central C32 — C3×C6 — C2×C12.59D6
 Upper central C1 — C2×C4 — C22×C4

Generators and relations for C2×C12.59D6
G = < a,b,c,d | a2=b12=c6=1, d2=b6, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b5, dcd-1=b6c-1 >

Subgroups: 1636 in 492 conjugacy classes, 165 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×6], C3 [×4], C4 [×4], C4 [×4], C22, C22 [×2], C22 [×10], S3 [×16], C6 [×12], C6 [×8], C2×C4 [×2], C2×C4 [×4], C2×C4 [×10], D4 [×12], Q8 [×4], C23, C23 [×2], C32, Dic3 [×16], C12 [×16], D6 [×32], C2×C6 [×12], C2×C6 [×8], C22×C4, C22×C4 [×2], C2×D4 [×3], C2×Q8, C4○D4 [×8], C3⋊S3 [×4], C3×C6, C3×C6 [×2], C3×C6 [×2], Dic6 [×16], C4×S3 [×32], D12 [×16], C2×Dic3 [×8], C3⋊D4 [×32], C2×C12 [×24], C22×S3 [×8], C22×C6 [×4], C2×C4○D4, C3⋊Dic3 [×4], C3×C12 [×4], C2×C3⋊S3 [×4], C2×C3⋊S3 [×4], C62, C62 [×2], C62 [×2], C2×Dic6 [×4], S3×C2×C4 [×8], C2×D12 [×4], C4○D12 [×32], C2×C3⋊D4 [×8], C22×C12 [×4], C324Q8 [×4], C4×C3⋊S3 [×8], C12⋊S3 [×4], C2×C3⋊Dic3 [×2], C327D4 [×8], C6×C12 [×2], C6×C12 [×4], C22×C3⋊S3 [×2], C2×C62, C2×C4○D12 [×4], C2×C324Q8, C2×C4×C3⋊S3 [×2], C2×C12⋊S3, C12.59D6 [×8], C2×C327D4 [×2], C2×C6×C12, C2×C12.59D6
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], C4○D12 [×8], S3×C23 [×4], C22×C3⋊S3 [×7], C2×C4○D12 [×4], C12.59D6 [×2], C23×C3⋊S3, C2×C12.59D6

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

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

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

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

84 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 ··· 6AB 12A ··· 12AF 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 12 ··· 12 size 1 1 1 1 2 2 18 18 18 18 2 2 2 2 1 1 1 1 2 2 18 18 18 18 2 ··· 2 2 ··· 2

84 irreducible representations

 dim 1 1 1 1 1 1 1 2 2 2 2 2 type + + + + + + + + + + image C1 C2 C2 C2 C2 C2 C2 S3 D6 D6 C4○D4 C4○D12 kernel C2×C12.59D6 C2×C32⋊4Q8 C2×C4×C3⋊S3 C2×C12⋊S3 C12.59D6 C2×C32⋊7D4 C2×C6×C12 C22×C12 C2×C12 C22×C6 C3×C6 C6 # reps 1 1 2 1 8 2 1 4 24 4 4 32

Matrix representation of C2×C12.59D6 in GL4(𝔽13) generated by

 12 0 0 0 0 12 0 0 0 0 12 0 0 0 0 12
,
 1 0 0 0 0 1 0 0 0 0 5 8 0 0 5 0
,
 1 1 0 0 12 0 0 0 0 0 4 11 0 0 2 2
,
 12 12 0 0 0 1 0 0 0 0 9 2 0 0 11 4
G:=sub<GL(4,GF(13))| [12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,5,5,0,0,8,0],[1,12,0,0,1,0,0,0,0,0,4,2,0,0,11,2],[12,0,0,0,12,1,0,0,0,0,9,11,0,0,2,4] >;

C2×C12.59D6 in GAP, Magma, Sage, TeX

C_2\times C_{12}._{59}D_6
% in TeX

G:=Group("C2xC12.59D6");
// GroupNames label

G:=SmallGroup(288,1006);
// by ID

G=gap.SmallGroup(288,1006);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,100,675,2693,9414]);
// 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,b*c=c*b,d*b*d^-1=b^5,d*c*d^-1=b^6*c^-1>;
// generators/relations

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