direct product, metabelian, nilpotent (class 2), monomial
Aliases: D4×C62, C62⋊7C23, C23⋊4C62, C4⋊(C2×C62), C24⋊7(C3×C6), (C2×C4)⋊4C62, C12⋊4(C22×C6), (C23×C6)⋊10C6, (C22×C12)⋊16C6, (C3×C12)⋊11C23, (C6×C12)⋊37C22, C6.21(C23×C6), (C3×C6).68C24, C22⋊2(C2×C62), (C22×C62)⋊2C2, (C2×C62)⋊14C22, C2.1(C22×C62), (C2×C6×C12)⋊19C2, (C2×C12)⋊15(C2×C6), (C22×C6)⋊8(C2×C6), (C2×C6)⋊4(C22×C6), (C22×C4)⋊7(C3×C6), SmallGroup(288,1019)
Series: Derived ►Chief ►Lower central ►Upper central
Subgroups: 948 in 708 conjugacy classes, 468 normal (10 characteristic)
C1, C2, C2 [×6], C2 [×8], C3 [×4], C4 [×4], C22 [×15], C22 [×24], C6 [×28], C6 [×32], C2×C4 [×6], D4 [×16], C23, C23 [×12], C23 [×8], C32, C12 [×16], C2×C6 [×60], C2×C6 [×96], C22×C4, C2×D4 [×12], C24 [×2], C3×C6, C3×C6 [×6], C3×C6 [×8], C2×C12 [×24], C3×D4 [×64], C22×C6 [×52], C22×C6 [×32], C22×D4, C3×C12 [×4], C62 [×15], C62 [×24], C22×C12 [×4], C6×D4 [×48], C23×C6 [×8], C6×C12 [×6], D4×C32 [×16], C2×C62, C2×C62 [×12], C2×C62 [×8], D4×C2×C6 [×4], C2×C6×C12, D4×C3×C6 [×12], C22×C62 [×2], D4×C62
Quotients:
C1, C2 [×15], C3 [×4], C22 [×35], C6 [×60], D4 [×4], C23 [×15], C32, C2×C6 [×140], C2×D4 [×6], C24, C3×C6 [×15], C3×D4 [×16], C22×C6 [×60], C22×D4, C62 [×35], C6×D4 [×24], C23×C6 [×4], D4×C32 [×4], C2×C62 [×15], D4×C2×C6 [×4], D4×C3×C6 [×6], C22×C62, D4×C62
Generators and relations
G = < a,b,c,d | a6=b6=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
►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 98 104 111 117 51)(2 99 105 112 118 52)(3 100 106 113 119 53)(4 101 107 114 120 54)(5 102 108 109 115 49)(6 97 103 110 116 50)(7 13 144 62 68 56)(8 14 139 63 69 57)(9 15 140 64 70 58)(10 16 141 65 71 59)(11 17 142 66 72 60)(12 18 143 61 67 55)(19 25 31 38 44 94)(20 26 32 39 45 95)(21 27 33 40 46 96)(22 28 34 41 47 91)(23 29 35 42 48 92)(24 30 36 37 43 93)(73 79 85 128 134 122)(74 80 86 129 135 123)(75 81 87 130 136 124)(76 82 88 131 137 125)(77 83 89 132 138 126)(78 84 90 127 133 121)
(1 7 42 74)(2 8 37 75)(3 9 38 76)(4 10 39 77)(5 11 40 78)(6 12 41 73)(13 48 80 98)(14 43 81 99)(15 44 82 100)(16 45 83 101)(17 46 84 102)(18 47 79 97)(19 131 113 64)(20 132 114 65)(21 127 109 66)(22 128 110 61)(23 129 111 62)(24 130 112 63)(25 137 119 70)(26 138 120 71)(27 133 115 72)(28 134 116 67)(29 135 117 68)(30 136 118 69)(31 125 53 58)(32 126 54 59)(33 121 49 60)(34 122 50 55)(35 123 51 56)(36 124 52 57)(85 103 143 91)(86 104 144 92)(87 105 139 93)(88 106 140 94)(89 107 141 95)(90 108 142 96)
(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), (1,98,104,111,117,51)(2,99,105,112,118,52)(3,100,106,113,119,53)(4,101,107,114,120,54)(5,102,108,109,115,49)(6,97,103,110,116,50)(7,13,144,62,68,56)(8,14,139,63,69,57)(9,15,140,64,70,58)(10,16,141,65,71,59)(11,17,142,66,72,60)(12,18,143,61,67,55)(19,25,31,38,44,94)(20,26,32,39,45,95)(21,27,33,40,46,96)(22,28,34,41,47,91)(23,29,35,42,48,92)(24,30,36,37,43,93)(73,79,85,128,134,122)(74,80,86,129,135,123)(75,81,87,130,136,124)(76,82,88,131,137,125)(77,83,89,132,138,126)(78,84,90,127,133,121), (1,7,42,74)(2,8,37,75)(3,9,38,76)(4,10,39,77)(5,11,40,78)(6,12,41,73)(13,48,80,98)(14,43,81,99)(15,44,82,100)(16,45,83,101)(17,46,84,102)(18,47,79,97)(19,131,113,64)(20,132,114,65)(21,127,109,66)(22,128,110,61)(23,129,111,62)(24,130,112,63)(25,137,119,70)(26,138,120,71)(27,133,115,72)(28,134,116,67)(29,135,117,68)(30,136,118,69)(31,125,53,58)(32,126,54,59)(33,121,49,60)(34,122,50,55)(35,123,51,56)(36,124,52,57)(85,103,143,91)(86,104,144,92)(87,105,139,93)(88,106,140,94)(89,107,141,95)(90,108,142,96), (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), (1,98,104,111,117,51)(2,99,105,112,118,52)(3,100,106,113,119,53)(4,101,107,114,120,54)(5,102,108,109,115,49)(6,97,103,110,116,50)(7,13,144,62,68,56)(8,14,139,63,69,57)(9,15,140,64,70,58)(10,16,141,65,71,59)(11,17,142,66,72,60)(12,18,143,61,67,55)(19,25,31,38,44,94)(20,26,32,39,45,95)(21,27,33,40,46,96)(22,28,34,41,47,91)(23,29,35,42,48,92)(24,30,36,37,43,93)(73,79,85,128,134,122)(74,80,86,129,135,123)(75,81,87,130,136,124)(76,82,88,131,137,125)(77,83,89,132,138,126)(78,84,90,127,133,121), (1,7,42,74)(2,8,37,75)(3,9,38,76)(4,10,39,77)(5,11,40,78)(6,12,41,73)(13,48,80,98)(14,43,81,99)(15,44,82,100)(16,45,83,101)(17,46,84,102)(18,47,79,97)(19,131,113,64)(20,132,114,65)(21,127,109,66)(22,128,110,61)(23,129,111,62)(24,130,112,63)(25,137,119,70)(26,138,120,71)(27,133,115,72)(28,134,116,67)(29,135,117,68)(30,136,118,69)(31,125,53,58)(32,126,54,59)(33,121,49,60)(34,122,50,55)(35,123,51,56)(36,124,52,57)(85,103,143,91)(86,104,144,92)(87,105,139,93)(88,106,140,94)(89,107,141,95)(90,108,142,96), (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)], [(1,98,104,111,117,51),(2,99,105,112,118,52),(3,100,106,113,119,53),(4,101,107,114,120,54),(5,102,108,109,115,49),(6,97,103,110,116,50),(7,13,144,62,68,56),(8,14,139,63,69,57),(9,15,140,64,70,58),(10,16,141,65,71,59),(11,17,142,66,72,60),(12,18,143,61,67,55),(19,25,31,38,44,94),(20,26,32,39,45,95),(21,27,33,40,46,96),(22,28,34,41,47,91),(23,29,35,42,48,92),(24,30,36,37,43,93),(73,79,85,128,134,122),(74,80,86,129,135,123),(75,81,87,130,136,124),(76,82,88,131,137,125),(77,83,89,132,138,126),(78,84,90,127,133,121)], [(1,7,42,74),(2,8,37,75),(3,9,38,76),(4,10,39,77),(5,11,40,78),(6,12,41,73),(13,48,80,98),(14,43,81,99),(15,44,82,100),(16,45,83,101),(17,46,84,102),(18,47,79,97),(19,131,113,64),(20,132,114,65),(21,127,109,66),(22,128,110,61),(23,129,111,62),(24,130,112,63),(25,137,119,70),(26,138,120,71),(27,133,115,72),(28,134,116,67),(29,135,117,68),(30,136,118,69),(31,125,53,58),(32,126,54,59),(33,121,49,60),(34,122,50,55),(35,123,51,56),(36,124,52,57),(85,103,143,91),(86,104,144,92),(87,105,139,93),(88,106,140,94),(89,107,141,95),(90,108,142,96)], [(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)])
Matrix representation ►G ⊆ 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] >;
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 |
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
×
⋊
ℤ
𝔽
○
≀