metabelian, supersoluble, monomial
Aliases: D4.(C3⋊Dic3), (C2×C12).160D6, C32⋊14(C8○D4), C62.65(C2×C4), (C3×Q8).9Dic3, (C3×D4).5Dic3, (D4×C32).3C4, (Q8×C32).3C4, Q8.2(C3⋊Dic3), C3⋊3(D4.Dic3), C12.21(C2×Dic3), C12.58D6⋊14C2, (C3×C12).182C23, C12.213(C22×S3), (C6×C12).152C22, C6.38(C22×Dic3), C32⋊4C8.41C22, C4.5(C2×C3⋊Dic3), (C3×C12).76(C2×C4), C4○D4.5(C3⋊S3), (C3×C4○D4).18S3, C4.42(C22×C3⋊S3), (C2×C32⋊4C8)⋊12C2, (C2×C6).10(C2×Dic3), (C32×C4○D4).5C2, C2.8(C22×C3⋊Dic3), C22.1(C2×C3⋊Dic3), (C3×C6).126(C22×C4), (C2×C4).58(C2×C3⋊S3), SmallGroup(288,805)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.(C3⋊Dic3)
G = < a,b,c,d,e | a4=b2=c3=1, d6=a2, e2=a2d3, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d5 >
Subgroups: 380 in 186 conjugacy classes, 113 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C6, C8, C2×C4, D4, Q8, C32, C12, C2×C6, C2×C8, M4(2), C4○D4, C3×C6, C3×C6, C3⋊C8, C2×C12, C3×D4, C3×Q8, C8○D4, C3×C12, C3×C12, C62, C2×C3⋊C8, C4.Dic3, C3×C4○D4, C32⋊4C8, C32⋊4C8, C6×C12, D4×C32, Q8×C32, D4.Dic3, C2×C32⋊4C8, C12.58D6, C32×C4○D4, D4.(C3⋊Dic3)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, Dic3, D6, C22×C4, C3⋊S3, C2×Dic3, C22×S3, C8○D4, C3⋊Dic3, C2×C3⋊S3, C22×Dic3, C2×C3⋊Dic3, C22×C3⋊S3, D4.Dic3, C22×C3⋊Dic3, D4.(C3⋊Dic3)
►On 144 points
Generators in S144
(1 31 7 25)(2 32 8 26)(3 33 9 27)(4 34 10 28)(5 35 11 29)(6 36 12 30)(13 51 19 57)(14 52 20 58)(15 53 21 59)(16 54 22 60)(17 55 23 49)(18 56 24 50)(37 75 43 81)(38 76 44 82)(39 77 45 83)(40 78 46 84)(41 79 47 73)(42 80 48 74)(61 121 67 127)(62 122 68 128)(63 123 69 129)(64 124 70 130)(65 125 71 131)(66 126 72 132)(85 114 91 120)(86 115 92 109)(87 116 93 110)(88 117 94 111)(89 118 95 112)(90 119 96 113)(97 136 103 142)(98 137 104 143)(99 138 105 144)(100 139 106 133)(101 140 107 134)(102 141 108 135)
(25 31)(26 32)(27 33)(28 34)(29 35)(30 36)(37 43)(38 44)(39 45)(40 46)(41 47)(42 48)(49 55)(50 56)(51 57)(52 58)(53 59)(54 60)(61 67)(62 68)(63 69)(64 70)(65 71)(66 72)(85 91)(86 92)(87 93)(88 94)(89 95)(90 96)(133 139)(134 140)(135 141)(136 142)(137 143)(138 144)
(1 75 118)(2 76 119)(3 77 120)(4 78 109)(5 79 110)(6 80 111)(7 81 112)(8 82 113)(9 83 114)(10 84 115)(11 73 116)(12 74 117)(13 107 122)(14 108 123)(15 97 124)(16 98 125)(17 99 126)(18 100 127)(19 101 128)(20 102 129)(21 103 130)(22 104 131)(23 105 132)(24 106 121)(25 37 89)(26 38 90)(27 39 91)(28 40 92)(29 41 93)(30 42 94)(31 43 95)(32 44 96)(33 45 85)(34 46 86)(35 47 87)(36 48 88)(49 144 66)(50 133 67)(51 134 68)(52 135 69)(53 136 70)(54 137 71)(55 138 72)(56 139 61)(57 140 62)(58 141 63)(59 142 64)(60 143 65)
(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)
(1 129 10 126 7 123 4 132)(2 122 11 131 8 128 5 125)(3 127 12 124 9 121 6 130)(13 116 22 113 19 110 16 119)(14 109 23 118 20 115 17 112)(15 114 24 111 21 120 18 117)(25 69 34 66 31 63 28 72)(26 62 35 71 32 68 29 65)(27 67 36 64 33 61 30 70)(37 135 46 144 43 141 40 138)(38 140 47 137 44 134 41 143)(39 133 48 142 45 139 42 136)(49 95 58 92 55 89 52 86)(50 88 59 85 56 94 53 91)(51 93 60 90 57 87 54 96)(73 104 82 101 79 98 76 107)(74 97 83 106 80 103 77 100)(75 102 84 99 81 108 78 105)
G:=sub<Sym(144)| (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,51,19,57)(14,52,20,58)(15,53,21,59)(16,54,22,60)(17,55,23,49)(18,56,24,50)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74)(61,121,67,127)(62,122,68,128)(63,123,69,129)(64,124,70,130)(65,125,71,131)(66,126,72,132)(85,114,91,120)(86,115,92,109)(87,116,93,110)(88,117,94,111)(89,118,95,112)(90,119,96,113)(97,136,103,142)(98,137,104,143)(99,138,105,144)(100,139,106,133)(101,140,107,134)(102,141,108,135), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144), (1,75,118)(2,76,119)(3,77,120)(4,78,109)(5,79,110)(6,80,111)(7,81,112)(8,82,113)(9,83,114)(10,84,115)(11,73,116)(12,74,117)(13,107,122)(14,108,123)(15,97,124)(16,98,125)(17,99,126)(18,100,127)(19,101,128)(20,102,129)(21,103,130)(22,104,131)(23,105,132)(24,106,121)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(33,45,85)(34,46,86)(35,47,87)(36,48,88)(49,144,66)(50,133,67)(51,134,68)(52,135,69)(53,136,70)(54,137,71)(55,138,72)(56,139,61)(57,140,62)(58,141,63)(59,142,64)(60,143,65), (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), (1,129,10,126,7,123,4,132)(2,122,11,131,8,128,5,125)(3,127,12,124,9,121,6,130)(13,116,22,113,19,110,16,119)(14,109,23,118,20,115,17,112)(15,114,24,111,21,120,18,117)(25,69,34,66,31,63,28,72)(26,62,35,71,32,68,29,65)(27,67,36,64,33,61,30,70)(37,135,46,144,43,141,40,138)(38,140,47,137,44,134,41,143)(39,133,48,142,45,139,42,136)(49,95,58,92,55,89,52,86)(50,88,59,85,56,94,53,91)(51,93,60,90,57,87,54,96)(73,104,82,101,79,98,76,107)(74,97,83,106,80,103,77,100)(75,102,84,99,81,108,78,105)>;
G:=Group( (1,31,7,25)(2,32,8,26)(3,33,9,27)(4,34,10,28)(5,35,11,29)(6,36,12,30)(13,51,19,57)(14,52,20,58)(15,53,21,59)(16,54,22,60)(17,55,23,49)(18,56,24,50)(37,75,43,81)(38,76,44,82)(39,77,45,83)(40,78,46,84)(41,79,47,73)(42,80,48,74)(61,121,67,127)(62,122,68,128)(63,123,69,129)(64,124,70,130)(65,125,71,131)(66,126,72,132)(85,114,91,120)(86,115,92,109)(87,116,93,110)(88,117,94,111)(89,118,95,112)(90,119,96,113)(97,136,103,142)(98,137,104,143)(99,138,105,144)(100,139,106,133)(101,140,107,134)(102,141,108,135), (25,31)(26,32)(27,33)(28,34)(29,35)(30,36)(37,43)(38,44)(39,45)(40,46)(41,47)(42,48)(49,55)(50,56)(51,57)(52,58)(53,59)(54,60)(61,67)(62,68)(63,69)(64,70)(65,71)(66,72)(85,91)(86,92)(87,93)(88,94)(89,95)(90,96)(133,139)(134,140)(135,141)(136,142)(137,143)(138,144), (1,75,118)(2,76,119)(3,77,120)(4,78,109)(5,79,110)(6,80,111)(7,81,112)(8,82,113)(9,83,114)(10,84,115)(11,73,116)(12,74,117)(13,107,122)(14,108,123)(15,97,124)(16,98,125)(17,99,126)(18,100,127)(19,101,128)(20,102,129)(21,103,130)(22,104,131)(23,105,132)(24,106,121)(25,37,89)(26,38,90)(27,39,91)(28,40,92)(29,41,93)(30,42,94)(31,43,95)(32,44,96)(33,45,85)(34,46,86)(35,47,87)(36,48,88)(49,144,66)(50,133,67)(51,134,68)(52,135,69)(53,136,70)(54,137,71)(55,138,72)(56,139,61)(57,140,62)(58,141,63)(59,142,64)(60,143,65), (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), (1,129,10,126,7,123,4,132)(2,122,11,131,8,128,5,125)(3,127,12,124,9,121,6,130)(13,116,22,113,19,110,16,119)(14,109,23,118,20,115,17,112)(15,114,24,111,21,120,18,117)(25,69,34,66,31,63,28,72)(26,62,35,71,32,68,29,65)(27,67,36,64,33,61,30,70)(37,135,46,144,43,141,40,138)(38,140,47,137,44,134,41,143)(39,133,48,142,45,139,42,136)(49,95,58,92,55,89,52,86)(50,88,59,85,56,94,53,91)(51,93,60,90,57,87,54,96)(73,104,82,101,79,98,76,107)(74,97,83,106,80,103,77,100)(75,102,84,99,81,108,78,105) );
G=PermutationGroup([[(1,31,7,25),(2,32,8,26),(3,33,9,27),(4,34,10,28),(5,35,11,29),(6,36,12,30),(13,51,19,57),(14,52,20,58),(15,53,21,59),(16,54,22,60),(17,55,23,49),(18,56,24,50),(37,75,43,81),(38,76,44,82),(39,77,45,83),(40,78,46,84),(41,79,47,73),(42,80,48,74),(61,121,67,127),(62,122,68,128),(63,123,69,129),(64,124,70,130),(65,125,71,131),(66,126,72,132),(85,114,91,120),(86,115,92,109),(87,116,93,110),(88,117,94,111),(89,118,95,112),(90,119,96,113),(97,136,103,142),(98,137,104,143),(99,138,105,144),(100,139,106,133),(101,140,107,134),(102,141,108,135)], [(25,31),(26,32),(27,33),(28,34),(29,35),(30,36),(37,43),(38,44),(39,45),(40,46),(41,47),(42,48),(49,55),(50,56),(51,57),(52,58),(53,59),(54,60),(61,67),(62,68),(63,69),(64,70),(65,71),(66,72),(85,91),(86,92),(87,93),(88,94),(89,95),(90,96),(133,139),(134,140),(135,141),(136,142),(137,143),(138,144)], [(1,75,118),(2,76,119),(3,77,120),(4,78,109),(5,79,110),(6,80,111),(7,81,112),(8,82,113),(9,83,114),(10,84,115),(11,73,116),(12,74,117),(13,107,122),(14,108,123),(15,97,124),(16,98,125),(17,99,126),(18,100,127),(19,101,128),(20,102,129),(21,103,130),(22,104,131),(23,105,132),(24,106,121),(25,37,89),(26,38,90),(27,39,91),(28,40,92),(29,41,93),(30,42,94),(31,43,95),(32,44,96),(33,45,85),(34,46,86),(35,47,87),(36,48,88),(49,144,66),(50,133,67),(51,134,68),(52,135,69),(53,136,70),(54,137,71),(55,138,72),(56,139,61),(57,140,62),(58,141,63),(59,142,64),(60,143,65)], [(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)], [(1,129,10,126,7,123,4,132),(2,122,11,131,8,128,5,125),(3,127,12,124,9,121,6,130),(13,116,22,113,19,110,16,119),(14,109,23,118,20,115,17,112),(15,114,24,111,21,120,18,117),(25,69,34,66,31,63,28,72),(26,62,35,71,32,68,29,65),(27,67,36,64,33,61,30,70),(37,135,46,144,43,141,40,138),(38,140,47,137,44,134,41,143),(39,133,48,142,45,139,42,136),(49,95,58,92,55,89,52,86),(50,88,59,85,56,94,53,91),(51,93,60,90,57,87,54,96),(73,104,82,101,79,98,76,107),(74,97,83,106,80,103,77,100),(75,102,84,99,81,108,78,105)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | ··· | 6P | 8A | 8B | 8C | 8D | 8E | ··· | 8J | 12A | ··· | 12H | 12I | ··· | 12T |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 9 | 9 | 9 | 9 | 18 | ··· | 18 | 2 | ··· | 2 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | - | - | ||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | Dic3 | Dic3 | C8○D4 | D4.Dic3 |
| kernel | D4.(C3⋊Dic3) | C2×C32⋊4C8 | C12.58D6 | C32×C4○D4 | D4×C32 | Q8×C32 | C3×C4○D4 | C2×C12 | C3×D4 | C3×Q8 | C32 | C3 |
| # reps | 1 | 3 | 3 | 1 | 6 | 2 | 4 | 12 | 12 | 4 | 4 | 8 |
Matrix representation of D4.(C3⋊Dic3) ►in GL6(𝔽73)
| 72 | 2 | 0 | 0 | 0 | 0 |
| 72 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 51 | 0 | 0 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 60 | 2 | 0 | 0 |
| 0 | 0 | 62 | 13 | 0 | 0 |
| 0 | 0 | 0 | 0 | 19 | 29 |
| 0 | 0 | 0 | 0 | 48 | 54 |
G:=sub<GL(6,GF(73))| [72,72,0,0,0,0,2,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,72,72,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[51,0,0,0,0,0,0,51,0,0,0,0,0,0,60,62,0,0,0,0,2,13,0,0,0,0,0,0,19,48,0,0,0,0,29,54] >;
D4.(C3⋊Dic3) in GAP, Magma, Sage, TeX
D_4.(C_3\rtimes {\rm Dic}_3) % in TeX
G:=Group("D4.(C3:Dic3)"); // GroupNames label
G:=SmallGroup(288,805);
// by ID
G=gap.SmallGroup(288,805);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,56,219,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^2=c^3=1,d^6=a^2,e^2=a^2*d^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^5>;
// generators/relations