direct product, metabelian, nilpotent (class 2), monomial
Aliases: D4×C3×C12, C122⋊18C2, C23.11C62, C62.287C23, C4⋊1(C6×C12), (C4×C12)⋊17C6, C12⋊6(C2×C12), C42⋊7(C3×C6), C6.83(C6×D4), C62⋊17(C2×C4), C22⋊2(C6×C12), (C22×C12)⋊8C6, (C6×D4).29C6, (C2×C4).20C62, C6.39(C22×C12), C22.7(C2×C62), (C6×C12).368C22, (C2×C62).87C22, (C2×C6×C12)⋊6C2, C4⋊C4⋊7(C3×C6), C2.3(D4×C3×C6), C2.4(C2×C6×C12), (C3×C4⋊C4)⋊16C6, (C2×C6)⋊7(C2×C12), (D4×C3×C6).21C2, (C3×C12)⋊21(C2×C4), C22⋊C4⋊6(C3×C6), (C22×C4)⋊4(C3×C6), (C2×D4).7(C3×C6), C6.49(C3×C4○D4), (C32×C4⋊C4)⋊25C2, (C3×C22⋊C4)⋊14C6, (C2×C12).95(C2×C6), (C3×C6).300(C2×D4), C2.2(C32×C4○D4), (C2×C6).93(C22×C6), (C22×C6).52(C2×C6), (C3×C6).166(C4○D4), (C32×C22⋊C4)⋊22C2, (C3×C6).131(C22×C4), SmallGroup(288,815)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C3×C12
G = < a,b,c,d | a3=b12=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 372 in 282 conjugacy classes, 192 normal (24 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, C2×C4, D4, C23, C32, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×C6, C3×C6, C2×C12, C2×C12, C3×D4, C22×C6, C4×D4, C3×C12, C3×C12, C62, C62, C62, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C22×C12, C6×D4, C6×C12, C6×C12, C6×C12, D4×C32, C2×C62, D4×C12, C122, C32×C22⋊C4, C32×C4⋊C4, C2×C6×C12, D4×C3×C6, D4×C3×C12
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C23, C32, C12, C2×C6, C22×C4, C2×D4, C4○D4, C3×C6, C2×C12, C3×D4, C22×C6, C4×D4, C3×C12, C62, C22×C12, C6×D4, C3×C4○D4, C6×C12, D4×C32, C2×C62, D4×C12, C2×C6×C12, D4×C3×C6, C32×C4○D4, D4×C3×C12
►On 144 points
Generators in S144
(1 130 119)(2 131 120)(3 132 109)(4 121 110)(5 122 111)(6 123 112)(7 124 113)(8 125 114)(9 126 115)(10 127 116)(11 128 117)(12 129 118)(13 97 51)(14 98 52)(15 99 53)(16 100 54)(17 101 55)(18 102 56)(19 103 57)(20 104 58)(21 105 59)(22 106 60)(23 107 49)(24 108 50)(25 144 80)(26 133 81)(27 134 82)(28 135 83)(29 136 84)(30 137 73)(31 138 74)(32 139 75)(33 140 76)(34 141 77)(35 142 78)(36 143 79)(37 92 70)(38 93 71)(39 94 72)(40 95 61)(41 96 62)(42 85 63)(43 86 64)(44 87 65)(45 88 66)(46 89 67)(47 90 68)(48 91 69)
(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 87 82 97)(2 88 83 98)(3 89 84 99)(4 90 73 100)(5 91 74 101)(6 92 75 102)(7 93 76 103)(8 94 77 104)(9 95 78 105)(10 96 79 106)(11 85 80 107)(12 86 81 108)(13 119 44 134)(14 120 45 135)(15 109 46 136)(16 110 47 137)(17 111 48 138)(18 112 37 139)(19 113 38 140)(20 114 39 141)(21 115 40 142)(22 116 41 143)(23 117 42 144)(24 118 43 133)(25 49 128 63)(26 50 129 64)(27 51 130 65)(28 52 131 66)(29 53 132 67)(30 54 121 68)(31 55 122 69)(32 56 123 70)(33 57 124 71)(34 58 125 72)(35 59 126 61)(36 60 127 62)
(1 103)(2 104)(3 105)(4 106)(5 107)(6 108)(7 97)(8 98)(9 99)(10 100)(11 101)(12 102)(13 113)(14 114)(15 115)(16 116)(17 117)(18 118)(19 119)(20 120)(21 109)(22 110)(23 111)(24 112)(25 69)(26 70)(27 71)(28 72)(29 61)(30 62)(31 63)(32 64)(33 65)(34 66)(35 67)(36 68)(37 133)(38 134)(39 135)(40 136)(41 137)(42 138)(43 139)(44 140)(45 141)(46 142)(47 143)(48 144)(49 122)(50 123)(51 124)(52 125)(53 126)(54 127)(55 128)(56 129)(57 130)(58 131)(59 132)(60 121)(73 96)(74 85)(75 86)(76 87)(77 88)(78 89)(79 90)(80 91)(81 92)(82 93)(83 94)(84 95)
G:=sub<Sym(144)| (1,130,119)(2,131,120)(3,132,109)(4,121,110)(5,122,111)(6,123,112)(7,124,113)(8,125,114)(9,126,115)(10,127,116)(11,128,117)(12,129,118)(13,97,51)(14,98,52)(15,99,53)(16,100,54)(17,101,55)(18,102,56)(19,103,57)(20,104,58)(21,105,59)(22,106,60)(23,107,49)(24,108,50)(25,144,80)(26,133,81)(27,134,82)(28,135,83)(29,136,84)(30,137,73)(31,138,74)(32,139,75)(33,140,76)(34,141,77)(35,142,78)(36,143,79)(37,92,70)(38,93,71)(39,94,72)(40,95,61)(41,96,62)(42,85,63)(43,86,64)(44,87,65)(45,88,66)(46,89,67)(47,90,68)(48,91,69), (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), (1,87,82,97)(2,88,83,98)(3,89,84,99)(4,90,73,100)(5,91,74,101)(6,92,75,102)(7,93,76,103)(8,94,77,104)(9,95,78,105)(10,96,79,106)(11,85,80,107)(12,86,81,108)(13,119,44,134)(14,120,45,135)(15,109,46,136)(16,110,47,137)(17,111,48,138)(18,112,37,139)(19,113,38,140)(20,114,39,141)(21,115,40,142)(22,116,41,143)(23,117,42,144)(24,118,43,133)(25,49,128,63)(26,50,129,64)(27,51,130,65)(28,52,131,66)(29,53,132,67)(30,54,121,68)(31,55,122,69)(32,56,123,70)(33,57,124,71)(34,58,125,72)(35,59,126,61)(36,60,127,62), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,109)(22,110)(23,111)(24,112)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95)>;
G:=Group( (1,130,119)(2,131,120)(3,132,109)(4,121,110)(5,122,111)(6,123,112)(7,124,113)(8,125,114)(9,126,115)(10,127,116)(11,128,117)(12,129,118)(13,97,51)(14,98,52)(15,99,53)(16,100,54)(17,101,55)(18,102,56)(19,103,57)(20,104,58)(21,105,59)(22,106,60)(23,107,49)(24,108,50)(25,144,80)(26,133,81)(27,134,82)(28,135,83)(29,136,84)(30,137,73)(31,138,74)(32,139,75)(33,140,76)(34,141,77)(35,142,78)(36,143,79)(37,92,70)(38,93,71)(39,94,72)(40,95,61)(41,96,62)(42,85,63)(43,86,64)(44,87,65)(45,88,66)(46,89,67)(47,90,68)(48,91,69), (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), (1,87,82,97)(2,88,83,98)(3,89,84,99)(4,90,73,100)(5,91,74,101)(6,92,75,102)(7,93,76,103)(8,94,77,104)(9,95,78,105)(10,96,79,106)(11,85,80,107)(12,86,81,108)(13,119,44,134)(14,120,45,135)(15,109,46,136)(16,110,47,137)(17,111,48,138)(18,112,37,139)(19,113,38,140)(20,114,39,141)(21,115,40,142)(22,116,41,143)(23,117,42,144)(24,118,43,133)(25,49,128,63)(26,50,129,64)(27,51,130,65)(28,52,131,66)(29,53,132,67)(30,54,121,68)(31,55,122,69)(32,56,123,70)(33,57,124,71)(34,58,125,72)(35,59,126,61)(36,60,127,62), (1,103)(2,104)(3,105)(4,106)(5,107)(6,108)(7,97)(8,98)(9,99)(10,100)(11,101)(12,102)(13,113)(14,114)(15,115)(16,116)(17,117)(18,118)(19,119)(20,120)(21,109)(22,110)(23,111)(24,112)(25,69)(26,70)(27,71)(28,72)(29,61)(30,62)(31,63)(32,64)(33,65)(34,66)(35,67)(36,68)(37,133)(38,134)(39,135)(40,136)(41,137)(42,138)(43,139)(44,140)(45,141)(46,142)(47,143)(48,144)(49,122)(50,123)(51,124)(52,125)(53,126)(54,127)(55,128)(56,129)(57,130)(58,131)(59,132)(60,121)(73,96)(74,85)(75,86)(76,87)(77,88)(78,89)(79,90)(80,91)(81,92)(82,93)(83,94)(84,95) );
G=PermutationGroup([[(1,130,119),(2,131,120),(3,132,109),(4,121,110),(5,122,111),(6,123,112),(7,124,113),(8,125,114),(9,126,115),(10,127,116),(11,128,117),(12,129,118),(13,97,51),(14,98,52),(15,99,53),(16,100,54),(17,101,55),(18,102,56),(19,103,57),(20,104,58),(21,105,59),(22,106,60),(23,107,49),(24,108,50),(25,144,80),(26,133,81),(27,134,82),(28,135,83),(29,136,84),(30,137,73),(31,138,74),(32,139,75),(33,140,76),(34,141,77),(35,142,78),(36,143,79),(37,92,70),(38,93,71),(39,94,72),(40,95,61),(41,96,62),(42,85,63),(43,86,64),(44,87,65),(45,88,66),(46,89,67),(47,90,68),(48,91,69)], [(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)], [(1,87,82,97),(2,88,83,98),(3,89,84,99),(4,90,73,100),(5,91,74,101),(6,92,75,102),(7,93,76,103),(8,94,77,104),(9,95,78,105),(10,96,79,106),(11,85,80,107),(12,86,81,108),(13,119,44,134),(14,120,45,135),(15,109,46,136),(16,110,47,137),(17,111,48,138),(18,112,37,139),(19,113,38,140),(20,114,39,141),(21,115,40,142),(22,116,41,143),(23,117,42,144),(24,118,43,133),(25,49,128,63),(26,50,129,64),(27,51,130,65),(28,52,131,66),(29,53,132,67),(30,54,121,68),(31,55,122,69),(32,56,123,70),(33,57,124,71),(34,58,125,72),(35,59,126,61),(36,60,127,62)], [(1,103),(2,104),(3,105),(4,106),(5,107),(6,108),(7,97),(8,98),(9,99),(10,100),(11,101),(12,102),(13,113),(14,114),(15,115),(16,116),(17,117),(18,118),(19,119),(20,120),(21,109),(22,110),(23,111),(24,112),(25,69),(26,70),(27,71),(28,72),(29,61),(30,62),(31,63),(32,64),(33,65),(34,66),(35,67),(36,68),(37,133),(38,134),(39,135),(40,136),(41,137),(42,138),(43,139),(44,140),(45,141),(46,142),(47,143),(48,144),(49,122),(50,123),(51,124),(52,125),(53,126),(54,127),(55,128),(56,129),(57,130),(58,131),(59,132),(60,121),(73,96),(74,85),(75,86),(76,87),(77,88),(78,89),(79,90),(80,91),(81,92),(82,93),(83,94),(84,95)]])
180 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | ··· | 3H | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6X | 6Y | ··· | 6BD | 12A | ··· | 12AF | 12AG | ··· | 12CR |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 | 1 | ··· | 1 | 2 | ··· | 2 |
180 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C12 | D4 | C4○D4 | C3×D4 | C3×C4○D4 |
| kernel | D4×C3×C12 | C122 | C32×C22⋊C4 | C32×C4⋊C4 | C2×C6×C12 | D4×C3×C6 | D4×C12 | D4×C32 | C4×C12 | C3×C22⋊C4 | C3×C4⋊C4 | C22×C12 | C6×D4 | C3×D4 | C3×C12 | C3×C6 | C12 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 8 | 8 | 16 | 8 | 16 | 8 | 64 | 2 | 2 | 16 | 16 |
Matrix representation of D4×C3×C12 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 5 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 8 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 11 | 1 |
| 0 | 0 | 8 | 2 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 11 | 11 |
| 0 | 0 | 8 | 2 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[5,0,0,0,0,4,0,0,0,0,8,0,0,0,0,8],[12,0,0,0,0,12,0,0,0,0,11,8,0,0,1,2],[12,0,0,0,0,1,0,0,0,0,11,8,0,0,11,2] >;
D4×C3×C12 in GAP, Magma, Sage, TeX
D_4\times C_3\times C_{12} % in TeX
G:=Group("D4xC3xC12"); // GroupNames label
G:=SmallGroup(288,815);
// by ID
G=gap.SmallGroup(288,815);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-3,-2,-2,1008,1037,772]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=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