direct product, metabelian, supersoluble, monomial
Aliases: C6×C32⋊4Q8, C62.152D6, C6⋊2(C3×Dic6), C3⋊3(C6×Dic6), (C3×C6)⋊8Dic6, C32⋊9(C6×Q8), (C32×C6)⋊7Q8, C33⋊17(C2×Q8), C12.83(S3×C6), (C6×C12).45S3, (C6×C12).29C6, (C3×C12).210D6, C62.75(C2×C6), C32⋊15(C2×Dic6), (C3×C62).61C22, (C32×C6).85C23, (C32×C12).86C22, C6.52(S3×C2×C6), (C3×C6)⋊6(C3×Q8), C4.11(C6×C3⋊S3), (C3×C6×C12).12C2, C12.87(C2×C3⋊S3), (C2×C6).75(S3×C6), C22.8(C6×C3⋊S3), (C3×C12).75(C2×C6), (C2×C12).21(C3×S3), C6.52(C22×C3⋊S3), (C2×C12).27(C3⋊S3), C3⋊Dic3.21(C2×C6), (C6×C3⋊Dic3).19C2, (C2×C3⋊Dic3).13C6, (C3×C6).59(C22×C6), (C3×C6).174(C22×S3), (C3×C3⋊Dic3).58C22, C2.3(C2×C6×C3⋊S3), (C2×C4).4(C3×C3⋊S3), (C2×C6).67(C2×C3⋊S3), SmallGroup(432,710)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C3 — C32 — C3×C6 — C32×C6 — C3×C3⋊Dic3 — C6×C3⋊Dic3 — C6×C32⋊4Q8 |
Generators and relations for C6×C32⋊4Q8
G = < a,b,c,d,e | a6=b3=c3=d4=1, e2=d2, ab=ba, ac=ca, ad=da, ae=ea, bc=cb, bd=db, ebe-1=b-1, cd=dc, ece-1=c-1, ede-1=d-1 >
Subgroups: 708 in 292 conjugacy classes, 118 normal (18 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, Q8, C32, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C2×Q8, C3×C6, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C2×C12, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C62, C2×Dic6, C6×Q8, C32×C6, C32×C6, C3×Dic6, C6×Dic3, C32⋊4Q8, C2×C3⋊Dic3, C6×C12, C6×C12, C6×C12, C3×C3⋊Dic3, C32×C12, C3×C62, C6×Dic6, C2×C32⋊4Q8, C3×C32⋊4Q8, C6×C3⋊Dic3, C3×C6×C12, C6×C32⋊4Q8
Quotients: C1, C2, C3, C22, S3, C6, Q8, C23, D6, C2×C6, C2×Q8, C3×S3, C3⋊S3, Dic6, C3×Q8, C22×S3, C22×C6, S3×C6, C2×C3⋊S3, C2×Dic6, C6×Q8, C3×C3⋊S3, C3×Dic6, C32⋊4Q8, S3×C2×C6, C22×C3⋊S3, C6×C3⋊S3, C6×Dic6, C2×C32⋊4Q8, C3×C32⋊4Q8, C2×C6×C3⋊S3, C6×C32⋊4Q8
►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 14 49)(2 15 50)(3 16 51)(4 17 52)(5 18 53)(6 13 54)(7 100 131)(8 101 132)(9 102 127)(10 97 128)(11 98 129)(12 99 130)(19 31 72)(20 32 67)(21 33 68)(22 34 69)(23 35 70)(24 36 71)(25 75 56)(26 76 57)(27 77 58)(28 78 59)(29 73 60)(30 74 55)(37 116 44)(38 117 45)(39 118 46)(40 119 47)(41 120 48)(42 115 43)(61 112 81)(62 113 82)(63 114 83)(64 109 84)(65 110 79)(66 111 80)(85 137 126)(86 138 121)(87 133 122)(88 134 123)(89 135 124)(90 136 125)(91 144 103)(92 139 104)(93 140 105)(94 141 106)(95 142 107)(96 143 108)
(1 16 53)(2 17 54)(3 18 49)(4 13 50)(5 14 51)(6 15 52)(7 98 127)(8 99 128)(9 100 129)(10 101 130)(11 102 131)(12 97 132)(19 33 70)(20 34 71)(21 35 72)(22 36 67)(23 31 68)(24 32 69)(25 77 60)(26 78 55)(27 73 56)(28 74 57)(29 75 58)(30 76 59)(37 120 46)(38 115 47)(39 116 48)(40 117 43)(41 118 44)(42 119 45)(61 114 79)(62 109 80)(63 110 81)(64 111 82)(65 112 83)(66 113 84)(85 135 122)(86 136 123)(87 137 124)(88 138 125)(89 133 126)(90 134 121)(91 142 105)(92 143 106)(93 144 107)(94 139 108)(95 140 103)(96 141 104)
(1 64 20 74)(2 65 21 75)(3 66 22 76)(4 61 23 77)(5 62 24 78)(6 63 19 73)(7 96 115 136)(8 91 116 137)(9 92 117 138)(10 93 118 133)(11 94 119 134)(12 95 120 135)(13 114 31 60)(14 109 32 55)(15 110 33 56)(16 111 34 57)(17 112 35 58)(18 113 36 59)(25 50 79 68)(26 51 80 69)(27 52 81 70)(28 53 82 71)(29 54 83 72)(30 49 84 67)(37 85 132 103)(38 86 127 104)(39 87 128 105)(40 88 129 106)(41 89 130 107)(42 90 131 108)(43 125 100 143)(44 126 101 144)(45 121 102 139)(46 122 97 140)(47 123 98 141)(48 124 99 142)
(1 95 20 135)(2 96 21 136)(3 91 22 137)(4 92 23 138)(5 93 24 133)(6 94 19 134)(7 75 115 65)(8 76 116 66)(9 77 117 61)(10 78 118 62)(11 73 119 63)(12 74 120 64)(13 106 31 88)(14 107 32 89)(15 108 33 90)(16 103 34 85)(17 104 35 86)(18 105 36 87)(25 43 79 100)(26 44 80 101)(27 45 81 102)(28 46 82 97)(29 47 83 98)(30 48 84 99)(37 111 132 57)(38 112 127 58)(39 113 128 59)(40 114 129 60)(41 109 130 55)(42 110 131 56)(49 142 67 124)(50 143 68 125)(51 144 69 126)(52 139 70 121)(53 140 71 122)(54 141 72 123)
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), (1,14,49)(2,15,50)(3,16,51)(4,17,52)(5,18,53)(6,13,54)(7,100,131)(8,101,132)(9,102,127)(10,97,128)(11,98,129)(12,99,130)(19,31,72)(20,32,67)(21,33,68)(22,34,69)(23,35,70)(24,36,71)(25,75,56)(26,76,57)(27,77,58)(28,78,59)(29,73,60)(30,74,55)(37,116,44)(38,117,45)(39,118,46)(40,119,47)(41,120,48)(42,115,43)(61,112,81)(62,113,82)(63,114,83)(64,109,84)(65,110,79)(66,111,80)(85,137,126)(86,138,121)(87,133,122)(88,134,123)(89,135,124)(90,136,125)(91,144,103)(92,139,104)(93,140,105)(94,141,106)(95,142,107)(96,143,108), (1,16,53)(2,17,54)(3,18,49)(4,13,50)(5,14,51)(6,15,52)(7,98,127)(8,99,128)(9,100,129)(10,101,130)(11,102,131)(12,97,132)(19,33,70)(20,34,71)(21,35,72)(22,36,67)(23,31,68)(24,32,69)(25,77,60)(26,78,55)(27,73,56)(28,74,57)(29,75,58)(30,76,59)(37,120,46)(38,115,47)(39,116,48)(40,117,43)(41,118,44)(42,119,45)(61,114,79)(62,109,80)(63,110,81)(64,111,82)(65,112,83)(66,113,84)(85,135,122)(86,136,123)(87,137,124)(88,138,125)(89,133,126)(90,134,121)(91,142,105)(92,143,106)(93,144,107)(94,139,108)(95,140,103)(96,141,104), (1,64,20,74)(2,65,21,75)(3,66,22,76)(4,61,23,77)(5,62,24,78)(6,63,19,73)(7,96,115,136)(8,91,116,137)(9,92,117,138)(10,93,118,133)(11,94,119,134)(12,95,120,135)(13,114,31,60)(14,109,32,55)(15,110,33,56)(16,111,34,57)(17,112,35,58)(18,113,36,59)(25,50,79,68)(26,51,80,69)(27,52,81,70)(28,53,82,71)(29,54,83,72)(30,49,84,67)(37,85,132,103)(38,86,127,104)(39,87,128,105)(40,88,129,106)(41,89,130,107)(42,90,131,108)(43,125,100,143)(44,126,101,144)(45,121,102,139)(46,122,97,140)(47,123,98,141)(48,124,99,142), (1,95,20,135)(2,96,21,136)(3,91,22,137)(4,92,23,138)(5,93,24,133)(6,94,19,134)(7,75,115,65)(8,76,116,66)(9,77,117,61)(10,78,118,62)(11,73,119,63)(12,74,120,64)(13,106,31,88)(14,107,32,89)(15,108,33,90)(16,103,34,85)(17,104,35,86)(18,105,36,87)(25,43,79,100)(26,44,80,101)(27,45,81,102)(28,46,82,97)(29,47,83,98)(30,48,84,99)(37,111,132,57)(38,112,127,58)(39,113,128,59)(40,114,129,60)(41,109,130,55)(42,110,131,56)(49,142,67,124)(50,143,68,125)(51,144,69,126)(52,139,70,121)(53,140,71,122)(54,141,72,123)>;
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), (1,14,49)(2,15,50)(3,16,51)(4,17,52)(5,18,53)(6,13,54)(7,100,131)(8,101,132)(9,102,127)(10,97,128)(11,98,129)(12,99,130)(19,31,72)(20,32,67)(21,33,68)(22,34,69)(23,35,70)(24,36,71)(25,75,56)(26,76,57)(27,77,58)(28,78,59)(29,73,60)(30,74,55)(37,116,44)(38,117,45)(39,118,46)(40,119,47)(41,120,48)(42,115,43)(61,112,81)(62,113,82)(63,114,83)(64,109,84)(65,110,79)(66,111,80)(85,137,126)(86,138,121)(87,133,122)(88,134,123)(89,135,124)(90,136,125)(91,144,103)(92,139,104)(93,140,105)(94,141,106)(95,142,107)(96,143,108), (1,16,53)(2,17,54)(3,18,49)(4,13,50)(5,14,51)(6,15,52)(7,98,127)(8,99,128)(9,100,129)(10,101,130)(11,102,131)(12,97,132)(19,33,70)(20,34,71)(21,35,72)(22,36,67)(23,31,68)(24,32,69)(25,77,60)(26,78,55)(27,73,56)(28,74,57)(29,75,58)(30,76,59)(37,120,46)(38,115,47)(39,116,48)(40,117,43)(41,118,44)(42,119,45)(61,114,79)(62,109,80)(63,110,81)(64,111,82)(65,112,83)(66,113,84)(85,135,122)(86,136,123)(87,137,124)(88,138,125)(89,133,126)(90,134,121)(91,142,105)(92,143,106)(93,144,107)(94,139,108)(95,140,103)(96,141,104), (1,64,20,74)(2,65,21,75)(3,66,22,76)(4,61,23,77)(5,62,24,78)(6,63,19,73)(7,96,115,136)(8,91,116,137)(9,92,117,138)(10,93,118,133)(11,94,119,134)(12,95,120,135)(13,114,31,60)(14,109,32,55)(15,110,33,56)(16,111,34,57)(17,112,35,58)(18,113,36,59)(25,50,79,68)(26,51,80,69)(27,52,81,70)(28,53,82,71)(29,54,83,72)(30,49,84,67)(37,85,132,103)(38,86,127,104)(39,87,128,105)(40,88,129,106)(41,89,130,107)(42,90,131,108)(43,125,100,143)(44,126,101,144)(45,121,102,139)(46,122,97,140)(47,123,98,141)(48,124,99,142), (1,95,20,135)(2,96,21,136)(3,91,22,137)(4,92,23,138)(5,93,24,133)(6,94,19,134)(7,75,115,65)(8,76,116,66)(9,77,117,61)(10,78,118,62)(11,73,119,63)(12,74,120,64)(13,106,31,88)(14,107,32,89)(15,108,33,90)(16,103,34,85)(17,104,35,86)(18,105,36,87)(25,43,79,100)(26,44,80,101)(27,45,81,102)(28,46,82,97)(29,47,83,98)(30,48,84,99)(37,111,132,57)(38,112,127,58)(39,113,128,59)(40,114,129,60)(41,109,130,55)(42,110,131,56)(49,142,67,124)(50,143,68,125)(51,144,69,126)(52,139,70,121)(53,140,71,122)(54,141,72,123) );
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)], [(1,14,49),(2,15,50),(3,16,51),(4,17,52),(5,18,53),(6,13,54),(7,100,131),(8,101,132),(9,102,127),(10,97,128),(11,98,129),(12,99,130),(19,31,72),(20,32,67),(21,33,68),(22,34,69),(23,35,70),(24,36,71),(25,75,56),(26,76,57),(27,77,58),(28,78,59),(29,73,60),(30,74,55),(37,116,44),(38,117,45),(39,118,46),(40,119,47),(41,120,48),(42,115,43),(61,112,81),(62,113,82),(63,114,83),(64,109,84),(65,110,79),(66,111,80),(85,137,126),(86,138,121),(87,133,122),(88,134,123),(89,135,124),(90,136,125),(91,144,103),(92,139,104),(93,140,105),(94,141,106),(95,142,107),(96,143,108)], [(1,16,53),(2,17,54),(3,18,49),(4,13,50),(5,14,51),(6,15,52),(7,98,127),(8,99,128),(9,100,129),(10,101,130),(11,102,131),(12,97,132),(19,33,70),(20,34,71),(21,35,72),(22,36,67),(23,31,68),(24,32,69),(25,77,60),(26,78,55),(27,73,56),(28,74,57),(29,75,58),(30,76,59),(37,120,46),(38,115,47),(39,116,48),(40,117,43),(41,118,44),(42,119,45),(61,114,79),(62,109,80),(63,110,81),(64,111,82),(65,112,83),(66,113,84),(85,135,122),(86,136,123),(87,137,124),(88,138,125),(89,133,126),(90,134,121),(91,142,105),(92,143,106),(93,144,107),(94,139,108),(95,140,103),(96,141,104)], [(1,64,20,74),(2,65,21,75),(3,66,22,76),(4,61,23,77),(5,62,24,78),(6,63,19,73),(7,96,115,136),(8,91,116,137),(9,92,117,138),(10,93,118,133),(11,94,119,134),(12,95,120,135),(13,114,31,60),(14,109,32,55),(15,110,33,56),(16,111,34,57),(17,112,35,58),(18,113,36,59),(25,50,79,68),(26,51,80,69),(27,52,81,70),(28,53,82,71),(29,54,83,72),(30,49,84,67),(37,85,132,103),(38,86,127,104),(39,87,128,105),(40,88,129,106),(41,89,130,107),(42,90,131,108),(43,125,100,143),(44,126,101,144),(45,121,102,139),(46,122,97,140),(47,123,98,141),(48,124,99,142)], [(1,95,20,135),(2,96,21,136),(3,91,22,137),(4,92,23,138),(5,93,24,133),(6,94,19,134),(7,75,115,65),(8,76,116,66),(9,77,117,61),(10,78,118,62),(11,73,119,63),(12,74,120,64),(13,106,31,88),(14,107,32,89),(15,108,33,90),(16,103,34,85),(17,104,35,86),(18,105,36,87),(25,43,79,100),(26,44,80,101),(27,45,81,102),(28,46,82,97),(29,47,83,98),(30,48,84,99),(37,111,132,57),(38,112,127,58),(39,113,128,59),(40,114,129,60),(41,109,130,55),(42,110,131,56),(49,142,67,124),(50,143,68,125),(51,144,69,126),(52,139,70,121),(53,140,71,122),(54,141,72,123)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6AP | 12A | ··· | 12AZ | 12BA | ··· | 12BH |
| 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 | 18 | 18 | 18 | 18 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 18 | ··· | 18 |
126 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | Q8 | D6 | D6 | C3×S3 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C3×Dic6 |
| kernel | C6×C32⋊4Q8 | C3×C32⋊4Q8 | C6×C3⋊Dic3 | C3×C6×C12 | C2×C32⋊4Q8 | C32⋊4Q8 | C2×C3⋊Dic3 | C6×C12 | C6×C12 | C32×C6 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 2 | 1 | 2 | 8 | 4 | 2 | 4 | 2 | 8 | 4 | 8 | 16 | 4 | 16 | 8 | 32 |
Matrix representation of C6×C32⋊4Q8 ►in GL4(𝔽13) generated by
| 10 | 0 | 0 | 0 |
| 0 | 10 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 9 | 0 | 0 | 0 |
| 7 | 3 | 0 | 0 |
| 0 | 0 | 12 | 1 |
| 0 | 0 | 12 | 0 |
| 8 | 0 | 0 | 0 |
| 10 | 5 | 0 | 0 |
| 0 | 0 | 10 | 6 |
| 0 | 0 | 7 | 3 |
| 8 | 8 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 8 |
| 0 | 0 | 0 | 8 |
G:=sub<GL(4,GF(13))| [10,0,0,0,0,10,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,1,0],[9,7,0,0,0,3,0,0,0,0,12,12,0,0,1,0],[8,10,0,0,0,5,0,0,0,0,10,7,0,0,6,3],[8,0,0,0,8,5,0,0,0,0,5,0,0,0,8,8] >;
C6×C32⋊4Q8 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes_4Q_8
% in TeX
G:=Group("C6xC3^2:4Q8"); // GroupNames label
G:=SmallGroup(432,710);
// by ID
G=gap.SmallGroup(432,710);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-3,-3,168,590,142,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^3=d^4=1,e^2=d^2,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,e*b*e^-1=b^-1,c*d=d*c,e*c*e^-1=c^-1,e*d*e^-1=d^-1>;
// generators/relations