direct product, metabelian, nilpotent (class 2), monomial
Aliases: D4×C62, C62⋊7C23, C23⋊4C62, C4⋊(C2×C62), (C2×C4)⋊4C62, C24⋊7(C3×C6), C12⋊4(C22×C6), (C23×C6)⋊10C6, (C3×C12)⋊11C23, (C22×C12)⋊16C6, (C6×C12)⋊37C22, (C3×C6).68C24, C6.21(C23×C6), (C22×C62)⋊2C2, C22⋊2(C2×C62), (C2×C62)⋊14C22, C2.1(C22×C62), (C2×C6×C12)⋊19C2, (C2×C12)⋊15(C2×C6), (C22×C4)⋊7(C3×C6), (C2×C6)⋊4(C22×C6), (C22×C6)⋊8(C2×C6), SmallGroup(288,1019)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C62
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 948 in 708 conjugacy classes, 468 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, D4, C23, C23, C23, C32, C12, C2×C6, C2×C6, C22×C4, C2×D4, C24, C3×C6, C3×C6, C3×C6, C2×C12, C3×D4, C22×C6, C22×C6, C22×D4, C3×C12, C62, C62, C22×C12, C6×D4, C23×C6, C6×C12, D4×C32, C2×C62, C2×C62, C2×C62, D4×C2×C6, C2×C6×C12, D4×C3×C6, C22×C62, D4×C62
Quotients: C1, C2, C3, C22, C6, D4, C23, C32, C2×C6, C2×D4, C24, C3×C6, C3×D4, C22×C6, C22×D4, C62, C6×D4, C23×C6, D4×C32, C2×C62, D4×C2×C6, D4×C3×C6, C22×C62, D4×C62
►On 144 points
Generators in S144
(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)
(1 106 115 100 109 51)(2 107 116 101 110 52)(3 108 117 102 111 53)(4 103 118 97 112 54)(5 104 119 98 113 49)(6 105 120 99 114 50)(7 140 64 60 13 72)(8 141 65 55 14 67)(9 142 66 56 15 68)(10 143 61 57 16 69)(11 144 62 58 17 70)(12 139 63 59 18 71)(19 33 48 27 42 94)(20 34 43 28 37 95)(21 35 44 29 38 96)(22 36 45 30 39 91)(23 31 46 25 40 92)(24 32 47 26 41 93)(73 87 138 81 132 122)(74 88 133 82 127 123)(75 89 134 83 128 124)(76 90 135 84 129 125)(77 85 136 79 130 126)(78 86 137 80 131 121)
(1 70 25 74)(2 71 26 75)(3 72 27 76)(4 67 28 77)(5 68 29 78)(6 69 30 73)(7 42 90 108)(8 37 85 103)(9 38 86 104)(10 39 87 105)(11 40 88 106)(12 41 89 107)(13 48 125 53)(14 43 126 54)(15 44 121 49)(16 45 122 50)(17 46 123 51)(18 47 124 52)(19 84 102 64)(20 79 97 65)(21 80 98 66)(22 81 99 61)(23 82 100 62)(24 83 101 63)(31 127 109 58)(32 128 110 59)(33 129 111 60)(34 130 112 55)(35 131 113 56)(36 132 114 57)(91 138 120 143)(92 133 115 144)(93 134 116 139)(94 135 117 140)(95 136 118 141)(96 137 119 142)
(1 74)(2 75)(3 76)(4 77)(5 78)(6 73)(7 42)(8 37)(9 38)(10 39)(11 40)(12 41)(13 48)(14 43)(15 44)(16 45)(17 46)(18 47)(19 64)(20 65)(21 66)(22 61)(23 62)(24 63)(25 70)(26 71)(27 72)(28 67)(29 68)(30 69)(31 58)(32 59)(33 60)(34 55)(35 56)(36 57)(49 121)(50 122)(51 123)(52 124)(53 125)(54 126)(79 97)(80 98)(81 99)(82 100)(83 101)(84 102)(85 103)(86 104)(87 105)(88 106)(89 107)(90 108)(91 143)(92 144)(93 139)(94 140)(95 141)(96 142)(109 127)(110 128)(111 129)(112 130)(113 131)(114 132)(115 133)(116 134)(117 135)(118 136)(119 137)(120 138)
G:=sub<Sym(144)| (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), (1,106,115,100,109,51)(2,107,116,101,110,52)(3,108,117,102,111,53)(4,103,118,97,112,54)(5,104,119,98,113,49)(6,105,120,99,114,50)(7,140,64,60,13,72)(8,141,65,55,14,67)(9,142,66,56,15,68)(10,143,61,57,16,69)(11,144,62,58,17,70)(12,139,63,59,18,71)(19,33,48,27,42,94)(20,34,43,28,37,95)(21,35,44,29,38,96)(22,36,45,30,39,91)(23,31,46,25,40,92)(24,32,47,26,41,93)(73,87,138,81,132,122)(74,88,133,82,127,123)(75,89,134,83,128,124)(76,90,135,84,129,125)(77,85,136,79,130,126)(78,86,137,80,131,121), (1,70,25,74)(2,71,26,75)(3,72,27,76)(4,67,28,77)(5,68,29,78)(6,69,30,73)(7,42,90,108)(8,37,85,103)(9,38,86,104)(10,39,87,105)(11,40,88,106)(12,41,89,107)(13,48,125,53)(14,43,126,54)(15,44,121,49)(16,45,122,50)(17,46,123,51)(18,47,124,52)(19,84,102,64)(20,79,97,65)(21,80,98,66)(22,81,99,61)(23,82,100,62)(24,83,101,63)(31,127,109,58)(32,128,110,59)(33,129,111,60)(34,130,112,55)(35,131,113,56)(36,132,114,57)(91,138,120,143)(92,133,115,144)(93,134,116,139)(94,135,117,140)(95,136,118,141)(96,137,119,142), (1,74)(2,75)(3,76)(4,77)(5,78)(6,73)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,48)(14,43)(15,44)(16,45)(17,46)(18,47)(19,64)(20,65)(21,66)(22,61)(23,62)(24,63)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,58)(32,59)(33,60)(34,55)(35,56)(36,57)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,143)(92,144)(93,139)(94,140)(95,141)(96,142)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138)>;
G:=Group( (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), (1,106,115,100,109,51)(2,107,116,101,110,52)(3,108,117,102,111,53)(4,103,118,97,112,54)(5,104,119,98,113,49)(6,105,120,99,114,50)(7,140,64,60,13,72)(8,141,65,55,14,67)(9,142,66,56,15,68)(10,143,61,57,16,69)(11,144,62,58,17,70)(12,139,63,59,18,71)(19,33,48,27,42,94)(20,34,43,28,37,95)(21,35,44,29,38,96)(22,36,45,30,39,91)(23,31,46,25,40,92)(24,32,47,26,41,93)(73,87,138,81,132,122)(74,88,133,82,127,123)(75,89,134,83,128,124)(76,90,135,84,129,125)(77,85,136,79,130,126)(78,86,137,80,131,121), (1,70,25,74)(2,71,26,75)(3,72,27,76)(4,67,28,77)(5,68,29,78)(6,69,30,73)(7,42,90,108)(8,37,85,103)(9,38,86,104)(10,39,87,105)(11,40,88,106)(12,41,89,107)(13,48,125,53)(14,43,126,54)(15,44,121,49)(16,45,122,50)(17,46,123,51)(18,47,124,52)(19,84,102,64)(20,79,97,65)(21,80,98,66)(22,81,99,61)(23,82,100,62)(24,83,101,63)(31,127,109,58)(32,128,110,59)(33,129,111,60)(34,130,112,55)(35,131,113,56)(36,132,114,57)(91,138,120,143)(92,133,115,144)(93,134,116,139)(94,135,117,140)(95,136,118,141)(96,137,119,142), (1,74)(2,75)(3,76)(4,77)(5,78)(6,73)(7,42)(8,37)(9,38)(10,39)(11,40)(12,41)(13,48)(14,43)(15,44)(16,45)(17,46)(18,47)(19,64)(20,65)(21,66)(22,61)(23,62)(24,63)(25,70)(26,71)(27,72)(28,67)(29,68)(30,69)(31,58)(32,59)(33,60)(34,55)(35,56)(36,57)(49,121)(50,122)(51,123)(52,124)(53,125)(54,126)(79,97)(80,98)(81,99)(82,100)(83,101)(84,102)(85,103)(86,104)(87,105)(88,106)(89,107)(90,108)(91,143)(92,144)(93,139)(94,140)(95,141)(96,142)(109,127)(110,128)(111,129)(112,130)(113,131)(114,132)(115,133)(116,134)(117,135)(118,136)(119,137)(120,138) );
G=PermutationGroup([[(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)], [(1,106,115,100,109,51),(2,107,116,101,110,52),(3,108,117,102,111,53),(4,103,118,97,112,54),(5,104,119,98,113,49),(6,105,120,99,114,50),(7,140,64,60,13,72),(8,141,65,55,14,67),(9,142,66,56,15,68),(10,143,61,57,16,69),(11,144,62,58,17,70),(12,139,63,59,18,71),(19,33,48,27,42,94),(20,34,43,28,37,95),(21,35,44,29,38,96),(22,36,45,30,39,91),(23,31,46,25,40,92),(24,32,47,26,41,93),(73,87,138,81,132,122),(74,88,133,82,127,123),(75,89,134,83,128,124),(76,90,135,84,129,125),(77,85,136,79,130,126),(78,86,137,80,131,121)], [(1,70,25,74),(2,71,26,75),(3,72,27,76),(4,67,28,77),(5,68,29,78),(6,69,30,73),(7,42,90,108),(8,37,85,103),(9,38,86,104),(10,39,87,105),(11,40,88,106),(12,41,89,107),(13,48,125,53),(14,43,126,54),(15,44,121,49),(16,45,122,50),(17,46,123,51),(18,47,124,52),(19,84,102,64),(20,79,97,65),(21,80,98,66),(22,81,99,61),(23,82,100,62),(24,83,101,63),(31,127,109,58),(32,128,110,59),(33,129,111,60),(34,130,112,55),(35,131,113,56),(36,132,114,57),(91,138,120,143),(92,133,115,144),(93,134,116,139),(94,135,117,140),(95,136,118,141),(96,137,119,142)], [(1,74),(2,75),(3,76),(4,77),(5,78),(6,73),(7,42),(8,37),(9,38),(10,39),(11,40),(12,41),(13,48),(14,43),(15,44),(16,45),(17,46),(18,47),(19,64),(20,65),(21,66),(22,61),(23,62),(24,63),(25,70),(26,71),(27,72),(28,67),(29,68),(30,69),(31,58),(32,59),(33,60),(34,55),(35,56),(36,57),(49,121),(50,122),(51,123),(52,124),(53,125),(54,126),(79,97),(80,98),(81,99),(82,100),(83,101),(84,102),(85,103),(86,104),(87,105),(88,106),(89,107),(90,108),(91,143),(92,144),(93,139),(94,140),(95,141),(96,142),(109,127),(110,128),(111,129),(112,130),(113,131),(114,132),(115,133),(116,134),(117,135),(118,136),(119,137),(120,138)]])
180 conjugacy classes
| class | 1 | 2A | ··· | 2G | 2H | ··· | 2O | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 6A | ··· | 6BD | 6BE | ··· | 6DP | 12A | ··· | 12AF |
| order | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 |
| kernel | D4×C62 | C2×C6×C12 | D4×C3×C6 | C22×C62 | D4×C2×C6 | C22×C12 | C6×D4 | C23×C6 | C62 | C2×C6 |
| # reps | 1 | 1 | 12 | 2 | 8 | 8 | 96 | 16 | 4 | 32 |
Matrix representation of D4×C62 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 4 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 12 |
| 0 | 0 | 2 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 11 | 12 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,10,0,0,0,0,9,0,0,0,0,9],[4,0,0,0,0,3,0,0,0,0,12,0,0,0,0,12],[12,0,0,0,0,12,0,0,0,0,12,2,0,0,12,1],[1,0,0,0,0,1,0,0,0,0,1,11,0,0,0,12] >;
D4×C62 in GAP, Magma, Sage, TeX
D_4\times C_6^2
% in TeX
G:=Group("D4xC6^2"); // GroupNames label
G:=SmallGroup(288,1019);
// by ID
G=gap.SmallGroup(288,1019);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-3,-2,2045]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^6=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations