direct product, metabelian, supersoluble, monomial
Aliases: C32×C4⋊Dic3, C62.164D6, (C3×C12)⋊8C12, C12⋊1(C3×C12), C6.8(C6×C12), C4⋊(C32×Dic3), C33⋊17(C4⋊C4), (C6×C12).37C6, (C6×C12).57S3, (C32×C12)⋊6C4, C12⋊3(C3×Dic3), C6.40(C3×D12), (C3×C6).87D12, C6.4(D4×C32), (C3×C12)⋊11Dic3, C6.2(Q8×C32), (C2×C6).16C62, C62.61(C2×C6), (C6×Dic3).7C6, C6.18(C3×Dic6), C6.38(C6×Dic3), (C3×C6).33Dic6, (C32×C6).55D4, C2.1(C32×D12), (C32×C6).11Q8, C2.2(C32×Dic6), (C3×C62).43C22, (C3×C6×C12).9C2, C3⋊2(C32×C4⋊C4), C22.5(S3×C3×C6), C2.4(Dic3×C3×C6), C32⋊10(C3×C4⋊C4), (C2×C12).8(C3×C6), (C2×C6).91(S3×C6), (C3×C6).47(C3×D4), (C3×C6).13(C3×Q8), (Dic3×C3×C6).6C2, (C2×C12).46(C3×S3), (C3×C6).59(C2×C12), (C2×C4).3(S3×C32), (C32×C6).66(C2×C4), (C2×Dic3).2(C3×C6), (C3×C6).79(C2×Dic3), SmallGroup(432,473)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×C4⋊Dic3
G = < a,b,c,d,e | a3=b3=c4=d6=1, e2=d3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 392 in 220 conjugacy classes, 114 normal (26 characteristic)
C1, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C4⋊C4, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, C2×C12, C33, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, C4⋊Dic3, C3×C4⋊C4, C32×C6, C6×Dic3, C6×C12, C6×C12, C6×C12, C32×Dic3, C32×C12, C3×C62, C3×C4⋊Dic3, C32×C4⋊C4, Dic3×C3×C6, C3×C6×C12, C32×C4⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C32, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, C3×C6, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C3×Dic3, C3×C12, S3×C6, C62, C4⋊Dic3, C3×C4⋊C4, S3×C32, C3×Dic6, C3×D12, C6×Dic3, C6×C12, D4×C32, Q8×C32, C32×Dic3, S3×C3×C6, C3×C4⋊Dic3, C32×C4⋊C4, C32×Dic6, C32×D12, Dic3×C3×C6, C32×C4⋊Dic3
►On 144 points
Generators in S144
(1 36 24)(2 31 19)(3 32 20)(4 33 21)(5 34 22)(6 35 23)(7 136 124)(8 137 125)(9 138 126)(10 133 121)(11 134 122)(12 135 123)(13 37 25)(14 38 26)(15 39 27)(16 40 28)(17 41 29)(18 42 30)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(49 74 61)(50 75 62)(51 76 63)(52 77 64)(53 78 65)(54 73 66)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(85 109 97)(86 110 98)(87 111 99)(88 112 100)(89 113 101)(90 114 102)(115 139 127)(116 140 128)(117 141 129)(118 142 130)(119 143 131)(120 144 132)
(1 5 3)(2 6 4)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 29 27)(26 30 28)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)(49 53 51)(50 54 52)(55 59 57)(56 60 58)(61 65 63)(62 66 64)(67 71 69)(68 72 70)(73 77 75)(74 78 76)(79 81 83)(80 82 84)(85 87 89)(86 88 90)(91 93 95)(92 94 96)(97 99 101)(98 100 102)(103 105 107)(104 106 108)(109 111 113)(110 112 114)(115 117 119)(116 118 120)(121 123 125)(122 124 126)(127 129 131)(128 130 132)(133 135 137)(134 136 138)(139 141 143)(140 142 144)
(1 53 17 48)(2 54 18 43)(3 49 13 44)(4 50 14 45)(5 51 15 46)(6 52 16 47)(7 108 144 112)(8 103 139 113)(9 104 140 114)(10 105 141 109)(11 106 142 110)(12 107 143 111)(19 66 30 55)(20 61 25 56)(21 62 26 57)(22 63 27 58)(23 64 28 59)(24 65 29 60)(31 73 42 67)(32 74 37 68)(33 75 38 69)(34 76 39 70)(35 77 40 71)(36 78 41 72)(79 115 89 125)(80 116 90 126)(81 117 85 121)(82 118 86 122)(83 119 87 123)(84 120 88 124)(91 127 101 137)(92 128 102 138)(93 129 97 133)(94 130 98 134)(95 131 99 135)(96 132 100 136)
(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 81 4 84)(2 80 5 83)(3 79 6 82)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 89 16 86)(14 88 17 85)(15 87 18 90)(19 92 22 95)(20 91 23 94)(21 96 24 93)(25 101 28 98)(26 100 29 97)(27 99 30 102)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 113 40 110)(38 112 41 109)(39 111 42 114)(43 116 46 119)(44 115 47 118)(45 120 48 117)(49 125 52 122)(50 124 53 121)(51 123 54 126)(55 128 58 131)(56 127 59 130)(57 132 60 129)(61 137 64 134)(62 136 65 133)(63 135 66 138)(67 140 70 143)(68 139 71 142)(69 144 72 141)
G:=sub<Sym(144)| (1,36,24)(2,31,19)(3,32,20)(4,33,21)(5,34,22)(6,35,23)(7,136,124)(8,137,125)(9,138,126)(10,133,121)(11,134,122)(12,135,123)(13,37,25)(14,38,26)(15,39,27)(16,40,28)(17,41,29)(18,42,30)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(49,74,61)(50,75,62)(51,76,63)(52,77,64)(53,78,65)(54,73,66)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(85,109,97)(86,110,98)(87,111,99)(88,112,100)(89,113,101)(90,114,102)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96)(97,99,101)(98,100,102)(103,105,107)(104,106,108)(109,111,113)(110,112,114)(115,117,119)(116,118,120)(121,123,125)(122,124,126)(127,129,131)(128,130,132)(133,135,137)(134,136,138)(139,141,143)(140,142,144), (1,53,17,48)(2,54,18,43)(3,49,13,44)(4,50,14,45)(5,51,15,46)(6,52,16,47)(7,108,144,112)(8,103,139,113)(9,104,140,114)(10,105,141,109)(11,106,142,110)(12,107,143,111)(19,66,30,55)(20,61,25,56)(21,62,26,57)(22,63,27,58)(23,64,28,59)(24,65,29,60)(31,73,42,67)(32,74,37,68)(33,75,38,69)(34,76,39,70)(35,77,40,71)(36,78,41,72)(79,115,89,125)(80,116,90,126)(81,117,85,121)(82,118,86,122)(83,119,87,123)(84,120,88,124)(91,127,101,137)(92,128,102,138)(93,129,97,133)(94,130,98,134)(95,131,99,135)(96,132,100,136), (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), (1,81,4,84)(2,80,5,83)(3,79,6,82)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,89,16,86)(14,88,17,85)(15,87,18,90)(19,92,22,95)(20,91,23,94)(21,96,24,93)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,116,46,119)(44,115,47,118)(45,120,48,117)(49,125,52,122)(50,124,53,121)(51,123,54,126)(55,128,58,131)(56,127,59,130)(57,132,60,129)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141)>;
G:=Group( (1,36,24)(2,31,19)(3,32,20)(4,33,21)(5,34,22)(6,35,23)(7,136,124)(8,137,125)(9,138,126)(10,133,121)(11,134,122)(12,135,123)(13,37,25)(14,38,26)(15,39,27)(16,40,28)(17,41,29)(18,42,30)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(49,74,61)(50,75,62)(51,76,63)(52,77,64)(53,78,65)(54,73,66)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(85,109,97)(86,110,98)(87,111,99)(88,112,100)(89,113,101)(90,114,102)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,5,3)(2,6,4)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,29,27)(26,30,28)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46)(49,53,51)(50,54,52)(55,59,57)(56,60,58)(61,65,63)(62,66,64)(67,71,69)(68,72,70)(73,77,75)(74,78,76)(79,81,83)(80,82,84)(85,87,89)(86,88,90)(91,93,95)(92,94,96)(97,99,101)(98,100,102)(103,105,107)(104,106,108)(109,111,113)(110,112,114)(115,117,119)(116,118,120)(121,123,125)(122,124,126)(127,129,131)(128,130,132)(133,135,137)(134,136,138)(139,141,143)(140,142,144), (1,53,17,48)(2,54,18,43)(3,49,13,44)(4,50,14,45)(5,51,15,46)(6,52,16,47)(7,108,144,112)(8,103,139,113)(9,104,140,114)(10,105,141,109)(11,106,142,110)(12,107,143,111)(19,66,30,55)(20,61,25,56)(21,62,26,57)(22,63,27,58)(23,64,28,59)(24,65,29,60)(31,73,42,67)(32,74,37,68)(33,75,38,69)(34,76,39,70)(35,77,40,71)(36,78,41,72)(79,115,89,125)(80,116,90,126)(81,117,85,121)(82,118,86,122)(83,119,87,123)(84,120,88,124)(91,127,101,137)(92,128,102,138)(93,129,97,133)(94,130,98,134)(95,131,99,135)(96,132,100,136), (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), (1,81,4,84)(2,80,5,83)(3,79,6,82)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,89,16,86)(14,88,17,85)(15,87,18,90)(19,92,22,95)(20,91,23,94)(21,96,24,93)(25,101,28,98)(26,100,29,97)(27,99,30,102)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,113,40,110)(38,112,41,109)(39,111,42,114)(43,116,46,119)(44,115,47,118)(45,120,48,117)(49,125,52,122)(50,124,53,121)(51,123,54,126)(55,128,58,131)(56,127,59,130)(57,132,60,129)(61,137,64,134)(62,136,65,133)(63,135,66,138)(67,140,70,143)(68,139,71,142)(69,144,72,141) );
G=PermutationGroup([[(1,36,24),(2,31,19),(3,32,20),(4,33,21),(5,34,22),(6,35,23),(7,136,124),(8,137,125),(9,138,126),(10,133,121),(11,134,122),(12,135,123),(13,37,25),(14,38,26),(15,39,27),(16,40,28),(17,41,29),(18,42,30),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60),(49,74,61),(50,75,62),(51,76,63),(52,77,64),(53,78,65),(54,73,66),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96),(85,109,97),(86,110,98),(87,111,99),(88,112,100),(89,113,101),(90,114,102),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,5,3),(2,6,4),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,29,27),(26,30,28),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46),(49,53,51),(50,54,52),(55,59,57),(56,60,58),(61,65,63),(62,66,64),(67,71,69),(68,72,70),(73,77,75),(74,78,76),(79,81,83),(80,82,84),(85,87,89),(86,88,90),(91,93,95),(92,94,96),(97,99,101),(98,100,102),(103,105,107),(104,106,108),(109,111,113),(110,112,114),(115,117,119),(116,118,120),(121,123,125),(122,124,126),(127,129,131),(128,130,132),(133,135,137),(134,136,138),(139,141,143),(140,142,144)], [(1,53,17,48),(2,54,18,43),(3,49,13,44),(4,50,14,45),(5,51,15,46),(6,52,16,47),(7,108,144,112),(8,103,139,113),(9,104,140,114),(10,105,141,109),(11,106,142,110),(12,107,143,111),(19,66,30,55),(20,61,25,56),(21,62,26,57),(22,63,27,58),(23,64,28,59),(24,65,29,60),(31,73,42,67),(32,74,37,68),(33,75,38,69),(34,76,39,70),(35,77,40,71),(36,78,41,72),(79,115,89,125),(80,116,90,126),(81,117,85,121),(82,118,86,122),(83,119,87,123),(84,120,88,124),(91,127,101,137),(92,128,102,138),(93,129,97,133),(94,130,98,134),(95,131,99,135),(96,132,100,136)], [(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)], [(1,81,4,84),(2,80,5,83),(3,79,6,82),(7,78,10,75),(8,77,11,74),(9,76,12,73),(13,89,16,86),(14,88,17,85),(15,87,18,90),(19,92,22,95),(20,91,23,94),(21,96,24,93),(25,101,28,98),(26,100,29,97),(27,99,30,102),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,113,40,110),(38,112,41,109),(39,111,42,114),(43,116,46,119),(44,115,47,118),(45,120,48,117),(49,125,52,122),(50,124,53,121),(51,123,54,126),(55,128,58,131),(56,127,59,130),(57,132,60,129),(61,137,64,134),(62,136,65,133),(63,135,66,138),(67,140,70,143),(68,139,71,142),(69,144,72,141)]])
162 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | ··· | 3H | 3I | ··· | 3Q | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6X | 6Y | ··· | 6AY | 12A | ··· | 12AZ | 12BA | ··· | 12CF |
| order | 1 | 2 | 2 | 2 | 3 | ··· | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 6 | ··· | 6 |
162 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | - | + | ||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | Dic6 | D12 | C3×D4 | C3×Q8 | C3×Dic3 | S3×C6 | C3×Dic6 | C3×D12 |
| kernel | C32×C4⋊Dic3 | Dic3×C3×C6 | C3×C6×C12 | C3×C4⋊Dic3 | C32×C12 | C6×Dic3 | C6×C12 | C3×C12 | C6×C12 | C32×C6 | C32×C6 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 1 | 1 | 1 | 2 | 1 | 8 | 2 | 2 | 8 | 8 | 16 | 8 | 16 | 16 |
Matrix representation of C32×C4⋊Dic3 ►in GL4(𝔽13) generated by
| 1 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 3 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 12 | 5 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 11 | 3 |
| 5 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 5 | 2 |
| 0 | 0 | 1 | 8 |
G:=sub<GL(4,GF(13))| [1,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[3,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[1,0,0,0,0,12,0,0,0,0,8,12,0,0,0,5],[12,0,0,0,0,1,0,0,0,0,9,11,0,0,0,3],[5,0,0,0,0,12,0,0,0,0,5,1,0,0,2,8] >;
C32×C4⋊Dic3 in GAP, Magma, Sage, TeX
C_3^2\times C_4\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3^2xC4:Dic3"); // GroupNames label
G:=SmallGroup(432,473);
// by ID
G=gap.SmallGroup(432,473);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,252,1037,512,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^4=d^6=1,e^2=d^3,a*b=b*a,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^-1>;
// generators/relations