direct product, metabelian, supersoluble, monomial
Aliases: C2×C12.59D6, C62.277C23, (C2×C12)⋊29D6, C6⋊5(C4○D12), (C6×C12)⋊32C22, (C22×C12)⋊12S3, C6.57(S3×C23), (C3×C6).56C24, (C22×C6).167D6, C12⋊S3⋊29C22, (C3×C12).184C23, C12.214(C22×S3), C32⋊7D4⋊16C22, C3⋊Dic3.46C23, C32⋊4Q8⋊27C22, (C2×C62).123C22, (C2×C6×C12)⋊10C2, C3⋊6(C2×C4○D12), (C3×C6)⋊8(C4○D4), C32⋊14(C2×C4○D4), C2.5(C23×C3⋊S3), (C4×C3⋊S3)⋊20C22, (C22×C4)⋊8(C3⋊S3), (C2×C12⋊S3)⋊23C2, C23.32(C2×C3⋊S3), C4.43(C22×C3⋊S3), (C2×C3⋊S3).50C23, (C2×C32⋊7D4)⋊21C2, (C2×C32⋊4Q8)⋊24C2, C22.5(C22×C3⋊S3), (C2×C6).286(C22×S3), (C22×C3⋊S3).106C22, (C2×C3⋊Dic3).182C22, (C2×C4×C3⋊S3)⋊26C2, (C2×C4)⋊10(C2×C3⋊S3), SmallGroup(288,1006)
Series: Derived ►Chief ►Lower central ►Upper central
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], C32⋊4Q8 [×4], C4×C3⋊S3 [×8], C12⋊S3 [×4], C2×C3⋊Dic3 [×2], C32⋊7D4 [×8], C6×C12 [×2], C6×C12 [×4], C22×C3⋊S3 [×2], C2×C62, C2×C4○D12 [×4], C2×C32⋊4Q8, C2×C4×C3⋊S3 [×2], C2×C12⋊S3, C12.59D6 [×8], C2×C32⋊7D4 [×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
►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