direct product, metabelian, supersoluble, monomial
Aliases: C32×Dic3⋊C4, C62.163D6, C6.4(C6×C12), Dic3⋊(C3×C12), C6.42(S3×C12), C33⋊16(C4⋊C4), (C6×C12).27S3, (C6×C12).25C6, C6.5(D4×C32), (C3×Dic3)⋊3C12, C6.1(Q8×C32), (C2×C6).15C62, C62.60(C2×C6), (C3×C6).32Dic6, C6.17(C3×Dic6), (C32×C6).65D4, (C32×C6).10Q8, (C6×Dic3).13C6, (C32×Dic3)⋊7C4, C2.1(C32×Dic6), (C3×C62).42C22, C2.4(S3×C3×C12), (C3×C6×C12).1C2, C3⋊1(C32×C4⋊C4), C32⋊9(C3×C4⋊C4), C22.4(S3×C3×C6), (C2×C12).9(C3×S3), (C3×C6).96(C4×S3), (C2×C6).90(S3×C6), (C2×C12).7(C3×C6), (C3×C6).61(C3×D4), C6.51(C3×C3⋊D4), (C3×C6).12(C3×Q8), (Dic3×C3×C6).5C2, (C3×C6).45(C2×C12), (C2×C4).1(S3×C32), C2.1(C32×C3⋊D4), (C32×C6).52(C2×C4), (C2×Dic3).1(C3×C6), (C3×C6).120(C3⋊D4), SmallGroup(432,472)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C32×Dic3⋊C4
G = < a,b,c,d,e | a3=b3=c6=e4=1, d2=c3, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd-1=c-1, ce=ec, ede-1=c3d >
Subgroups: 392 in 220 conjugacy classes, 102 normal (30 characteristic)
C1, C2, C3, C3, C3, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, C32, Dic3, Dic3, 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×Dic3, C3×C12, C62, C62, C62, Dic3⋊C4, C3×C4⋊C4, C32×C6, C6×Dic3, C6×C12, C6×C12, C6×C12, C32×Dic3, C32×Dic3, C32×C12, C3×C62, C3×Dic3⋊C4, C32×C4⋊C4, Dic3×C3×C6, C3×C6×C12, C32×Dic3⋊C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, C32, C12, D6, C2×C6, C4⋊C4, C3×S3, C3×C6, Dic6, C4×S3, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×C12, S3×C6, C62, Dic3⋊C4, C3×C4⋊C4, S3×C32, C3×Dic6, S3×C12, C3×C3⋊D4, C6×C12, D4×C32, Q8×C32, S3×C3×C6, C3×Dic3⋊C4, C32×C4⋊C4, C32×Dic6, S3×C3×C12, C32×C3⋊D4, C32×Dic3⋊C4
►On 144 points
Generators in S144
(1 25 13)(2 26 14)(3 27 15)(4 28 16)(5 29 17)(6 30 18)(7 31 19)(8 32 20)(9 33 21)(10 34 22)(11 35 23)(12 36 24)(37 61 49)(38 62 50)(39 63 51)(40 64 52)(41 65 53)(42 66 54)(43 67 55)(44 68 56)(45 69 57)(46 70 58)(47 71 59)(48 72 60)(73 97 85)(74 98 86)(75 99 87)(76 100 88)(77 101 89)(78 102 90)(79 103 91)(80 104 92)(81 105 93)(82 106 94)(83 107 95)(84 108 96)(109 133 121)(110 134 122)(111 135 123)(112 136 124)(113 137 125)(114 138 126)(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 11 9)(8 12 10)(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 75 77)(74 76 78)(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 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 80 4 83)(2 79 5 82)(3 84 6 81)(7 74 10 77)(8 73 11 76)(9 78 12 75)(13 92 16 95)(14 91 17 94)(15 96 18 93)(19 86 22 89)(20 85 23 88)(21 90 24 87)(25 104 28 107)(26 103 29 106)(27 108 30 105)(31 98 34 101)(32 97 35 100)(33 102 36 99)(37 116 40 119)(38 115 41 118)(39 120 42 117)(43 110 46 113)(44 109 47 112)(45 114 48 111)(49 128 52 131)(50 127 53 130)(51 132 54 129)(55 122 58 125)(56 121 59 124)(57 126 60 123)(61 140 64 143)(62 139 65 142)(63 144 66 141)(67 134 70 137)(68 133 71 136)(69 138 72 135)
(1 43 7 37)(2 44 8 38)(3 45 9 39)(4 46 10 40)(5 47 11 41)(6 48 12 42)(13 55 19 49)(14 56 20 50)(15 57 21 51)(16 58 22 52)(17 59 23 53)(18 60 24 54)(25 67 31 61)(26 68 32 62)(27 69 33 63)(28 70 34 64)(29 71 35 65)(30 72 36 66)(73 118 79 112)(74 119 80 113)(75 120 81 114)(76 115 82 109)(77 116 83 110)(78 117 84 111)(85 130 91 124)(86 131 92 125)(87 132 93 126)(88 127 94 121)(89 128 95 122)(90 129 96 123)(97 142 103 136)(98 143 104 137)(99 144 105 138)(100 139 106 133)(101 140 107 134)(102 141 108 135)
G:=sub<Sym(144)| (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(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)(73,75,77)(74,76,78)(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), (1,2,3,4,5,6)(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), (1,80,4,83)(2,79,5,82)(3,84,6,81)(7,74,10,77)(8,73,11,76)(9,78,12,75)(13,92,16,95)(14,91,17,94)(15,96,18,93)(19,86,22,89)(20,85,23,88)(21,90,24,87)(25,104,28,107)(26,103,29,106)(27,108,30,105)(31,98,34,101)(32,97,35,100)(33,102,36,99)(37,116,40,119)(38,115,41,118)(39,120,42,117)(43,110,46,113)(44,109,47,112)(45,114,48,111)(49,128,52,131)(50,127,53,130)(51,132,54,129)(55,122,58,125)(56,121,59,124)(57,126,60,123)(61,140,64,143)(62,139,65,142)(63,144,66,141)(67,134,70,137)(68,133,71,136)(69,138,72,135), (1,43,7,37)(2,44,8,38)(3,45,9,39)(4,46,10,40)(5,47,11,41)(6,48,12,42)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(73,118,79,112)(74,119,80,113)(75,120,81,114)(76,115,82,109)(77,116,83,110)(78,117,84,111)(85,130,91,124)(86,131,92,125)(87,132,93,126)(88,127,94,121)(89,128,95,122)(90,129,96,123)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,139,106,133)(101,140,107,134)(102,141,108,135)>;
G:=Group( (1,25,13)(2,26,14)(3,27,15)(4,28,16)(5,29,17)(6,30,18)(7,31,19)(8,32,20)(9,33,21)(10,34,22)(11,35,23)(12,36,24)(37,61,49)(38,62,50)(39,63,51)(40,64,52)(41,65,53)(42,66,54)(43,67,55)(44,68,56)(45,69,57)(46,70,58)(47,71,59)(48,72,60)(73,97,85)(74,98,86)(75,99,87)(76,100,88)(77,101,89)(78,102,90)(79,103,91)(80,104,92)(81,105,93)(82,106,94)(83,107,95)(84,108,96)(109,133,121)(110,134,122)(111,135,123)(112,136,124)(113,137,125)(114,138,126)(115,139,127)(116,140,128)(117,141,129)(118,142,130)(119,143,131)(120,144,132), (1,5,3)(2,6,4)(7,11,9)(8,12,10)(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)(73,75,77)(74,76,78)(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), (1,2,3,4,5,6)(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), (1,80,4,83)(2,79,5,82)(3,84,6,81)(7,74,10,77)(8,73,11,76)(9,78,12,75)(13,92,16,95)(14,91,17,94)(15,96,18,93)(19,86,22,89)(20,85,23,88)(21,90,24,87)(25,104,28,107)(26,103,29,106)(27,108,30,105)(31,98,34,101)(32,97,35,100)(33,102,36,99)(37,116,40,119)(38,115,41,118)(39,120,42,117)(43,110,46,113)(44,109,47,112)(45,114,48,111)(49,128,52,131)(50,127,53,130)(51,132,54,129)(55,122,58,125)(56,121,59,124)(57,126,60,123)(61,140,64,143)(62,139,65,142)(63,144,66,141)(67,134,70,137)(68,133,71,136)(69,138,72,135), (1,43,7,37)(2,44,8,38)(3,45,9,39)(4,46,10,40)(5,47,11,41)(6,48,12,42)(13,55,19,49)(14,56,20,50)(15,57,21,51)(16,58,22,52)(17,59,23,53)(18,60,24,54)(25,67,31,61)(26,68,32,62)(27,69,33,63)(28,70,34,64)(29,71,35,65)(30,72,36,66)(73,118,79,112)(74,119,80,113)(75,120,81,114)(76,115,82,109)(77,116,83,110)(78,117,84,111)(85,130,91,124)(86,131,92,125)(87,132,93,126)(88,127,94,121)(89,128,95,122)(90,129,96,123)(97,142,103,136)(98,143,104,137)(99,144,105,138)(100,139,106,133)(101,140,107,134)(102,141,108,135) );
G=PermutationGroup([[(1,25,13),(2,26,14),(3,27,15),(4,28,16),(5,29,17),(6,30,18),(7,31,19),(8,32,20),(9,33,21),(10,34,22),(11,35,23),(12,36,24),(37,61,49),(38,62,50),(39,63,51),(40,64,52),(41,65,53),(42,66,54),(43,67,55),(44,68,56),(45,69,57),(46,70,58),(47,71,59),(48,72,60),(73,97,85),(74,98,86),(75,99,87),(76,100,88),(77,101,89),(78,102,90),(79,103,91),(80,104,92),(81,105,93),(82,106,94),(83,107,95),(84,108,96),(109,133,121),(110,134,122),(111,135,123),(112,136,124),(113,137,125),(114,138,126),(115,139,127),(116,140,128),(117,141,129),(118,142,130),(119,143,131),(120,144,132)], [(1,5,3),(2,6,4),(7,11,9),(8,12,10),(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),(73,75,77),(74,76,78),(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)], [(1,2,3,4,5,6),(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)], [(1,80,4,83),(2,79,5,82),(3,84,6,81),(7,74,10,77),(8,73,11,76),(9,78,12,75),(13,92,16,95),(14,91,17,94),(15,96,18,93),(19,86,22,89),(20,85,23,88),(21,90,24,87),(25,104,28,107),(26,103,29,106),(27,108,30,105),(31,98,34,101),(32,97,35,100),(33,102,36,99),(37,116,40,119),(38,115,41,118),(39,120,42,117),(43,110,46,113),(44,109,47,112),(45,114,48,111),(49,128,52,131),(50,127,53,130),(51,132,54,129),(55,122,58,125),(56,121,59,124),(57,126,60,123),(61,140,64,143),(62,139,65,142),(63,144,66,141),(67,134,70,137),(68,133,71,136),(69,138,72,135)], [(1,43,7,37),(2,44,8,38),(3,45,9,39),(4,46,10,40),(5,47,11,41),(6,48,12,42),(13,55,19,49),(14,56,20,50),(15,57,21,51),(16,58,22,52),(17,59,23,53),(18,60,24,54),(25,67,31,61),(26,68,32,62),(27,69,33,63),(28,70,34,64),(29,71,35,65),(30,72,36,66),(73,118,79,112),(74,119,80,113),(75,120,81,114),(76,115,82,109),(77,116,83,110),(78,117,84,111),(85,130,91,124),(86,131,92,125),(87,132,93,126),(88,127,94,121),(89,128,95,122),(90,129,96,123),(97,142,103,136),(98,143,104,137),(99,144,105,138),(100,139,106,133),(101,140,107,134),(102,141,108,135)]])
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 | D6 | C3×S3 | Dic6 | C4×S3 | C3⋊D4 | C3×D4 | C3×Q8 | S3×C6 | C3×Dic6 | S3×C12 | C3×C3⋊D4 |
| kernel | C32×Dic3⋊C4 | Dic3×C3×C6 | C3×C6×C12 | C3×Dic3⋊C4 | C32×Dic3 | C6×Dic3 | C6×C12 | C3×Dic3 | C6×C12 | C32×C6 | C32×C6 | C62 | C2×C12 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C3×C6 | C2×C6 | C6 | C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 16 | 8 | 32 | 1 | 1 | 1 | 1 | 8 | 2 | 2 | 2 | 8 | 8 | 8 | 16 | 16 | 16 |
Matrix representation of C32×Dic3⋊C4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 0 | 3 |
| 9 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 4 |
| 0 | 12 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 2 | 12 |
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,1,0,0,0,0,1],[9,0,0,0,0,9,0,0,0,0,3,0,0,0,0,3],[9,0,0,0,0,3,0,0,0,0,10,0,0,0,10,4],[0,12,0,0,12,0,0,0,0,0,1,2,0,0,12,12],[5,0,0,0,0,5,0,0,0,0,5,0,0,0,8,8] >;
C32×Dic3⋊C4 in GAP, Magma, Sage, TeX
C_3^2\times {\rm Dic}_3\rtimes C_4 % in TeX
G:=Group("C3^2xDic3:C4"); // GroupNames label
G:=SmallGroup(432,472);
// by ID
G=gap.SmallGroup(432,472);
# by ID
G:=PCGroup([7,-2,-2,-3,-3,-2,-2,-3,504,1037,260,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^6=e^4=1,d^2=c^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,d*c*d^-1=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations