direct product, metabelian, supersoluble, monomial, A-group
Aliases: C6×C32⋊4C8, C62.23C12, C62.23Dic3, (C3×C6)⋊6C24, (C32×C6)⋊6C8, C33⋊20(C2×C8), (C6×C12).44C6, (C6×C12).52S3, (C3×C62).8C4, C12.103(S3×C6), (C3×C12).25C12, C32⋊12(C2×C24), (C3×C12).225D6, C6.24(C6×Dic3), (C32×C12).14C4, C12.18(C3×Dic3), (C3×C12).29Dic3, C12.16(C3⋊Dic3), (C32×C12).93C22, C6⋊(C3×C3⋊C8), C3⋊2(C6×C3⋊C8), (C3×C6)⋊5(C3⋊C8), C4.14(C6×C3⋊S3), C32⋊11(C2×C3⋊C8), (C3×C6×C12).15C2, C12.96(C2×C3⋊S3), C4.3(C3×C3⋊Dic3), C2.1(C6×C3⋊Dic3), (C3×C6).61(C2×C12), (C3×C12).98(C2×C6), (C2×C12).38(C3×S3), C6.20(C2×C3⋊Dic3), (C2×C12).31(C3⋊S3), (C32×C6).68(C2×C4), (C3×C6).66(C2×Dic3), (C2×C6).24(C3×Dic3), C22.2(C3×C3⋊Dic3), (C2×C6).15(C3⋊Dic3), (C2×C4).5(C3×C3⋊S3), SmallGroup(432,485)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C6×C32⋊4C8 |
Generators and relations for C6×C32⋊4C8
G = < a,b,c,d | a6=b3=c3=d8=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 356 in 196 conjugacy classes, 102 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C22, C6, C6, C6, C8, C2×C4, C32, C32, C32, C12, C12, C12, C2×C6, C2×C6, C2×C6, C2×C8, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C2×C12, C2×C12, C2×C12, C33, C3×C12, C3×C12, C3×C12, C62, C62, C62, C2×C3⋊C8, C2×C24, C32×C6, C32×C6, C3×C3⋊C8, C32⋊4C8, C6×C12, C6×C12, C6×C12, C32×C12, C3×C62, C6×C3⋊C8, C2×C32⋊4C8, C3×C32⋊4C8, C3×C6×C12, C6×C32⋊4C8
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, Dic3, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3⋊C8, C24, C2×Dic3, C2×C12, C3×Dic3, C3⋊Dic3, S3×C6, C2×C3⋊S3, C2×C3⋊C8, C2×C24, C3×C3⋊S3, C3×C3⋊C8, C32⋊4C8, C6×Dic3, C2×C3⋊Dic3, C3×C3⋊Dic3, C6×C3⋊S3, C6×C3⋊C8, C2×C32⋊4C8, C3×C32⋊4C8, C6×C3⋊Dic3, C6×C32⋊4C8
►On 144 points
Generators in S144
(1 37 52 66 18 122)(2 38 53 67 19 123)(3 39 54 68 20 124)(4 40 55 69 21 125)(5 33 56 70 22 126)(6 34 49 71 23 127)(7 35 50 72 24 128)(8 36 51 65 17 121)(9 77 133 47 29 100)(10 78 134 48 30 101)(11 79 135 41 31 102)(12 80 136 42 32 103)(13 73 129 43 25 104)(14 74 130 44 26 97)(15 75 131 45 27 98)(16 76 132 46 28 99)(57 119 108 143 95 84)(58 120 109 144 96 85)(59 113 110 137 89 86)(60 114 111 138 90 87)(61 115 112 139 91 88)(62 116 105 140 92 81)(63 117 106 141 93 82)(64 118 107 142 94 83)
(1 63 11)(2 12 64)(3 57 13)(4 14 58)(5 59 15)(6 16 60)(7 61 9)(8 10 62)(17 30 92)(18 93 31)(19 32 94)(20 95 25)(21 26 96)(22 89 27)(23 28 90)(24 91 29)(33 113 75)(34 76 114)(35 115 77)(36 78 116)(37 117 79)(38 80 118)(39 119 73)(40 74 120)(41 66 141)(42 142 67)(43 68 143)(44 144 69)(45 70 137)(46 138 71)(47 72 139)(48 140 65)(49 132 111)(50 112 133)(51 134 105)(52 106 135)(53 136 107)(54 108 129)(55 130 109)(56 110 131)(81 121 101)(82 102 122)(83 123 103)(84 104 124)(85 125 97)(86 98 126)(87 127 99)(88 100 128)
(1 106 31)(2 32 107)(3 108 25)(4 26 109)(5 110 27)(6 28 111)(7 112 29)(8 30 105)(9 50 91)(10 92 51)(11 52 93)(12 94 53)(13 54 95)(14 96 55)(15 56 89)(16 90 49)(17 134 62)(18 63 135)(19 136 64)(20 57 129)(21 130 58)(22 59 131)(23 132 60)(24 61 133)(33 137 98)(34 99 138)(35 139 100)(36 101 140)(37 141 102)(38 103 142)(39 143 104)(40 97 144)(41 122 117)(42 118 123)(43 124 119)(44 120 125)(45 126 113)(46 114 127)(47 128 115)(48 116 121)(65 78 81)(66 82 79)(67 80 83)(68 84 73)(69 74 85)(70 86 75)(71 76 87)(72 88 77)
(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)
G:=sub<Sym(144)| (1,37,52,66,18,122)(2,38,53,67,19,123)(3,39,54,68,20,124)(4,40,55,69,21,125)(5,33,56,70,22,126)(6,34,49,71,23,127)(7,35,50,72,24,128)(8,36,51,65,17,121)(9,77,133,47,29,100)(10,78,134,48,30,101)(11,79,135,41,31,102)(12,80,136,42,32,103)(13,73,129,43,25,104)(14,74,130,44,26,97)(15,75,131,45,27,98)(16,76,132,46,28,99)(57,119,108,143,95,84)(58,120,109,144,96,85)(59,113,110,137,89,86)(60,114,111,138,90,87)(61,115,112,139,91,88)(62,116,105,140,92,81)(63,117,106,141,93,82)(64,118,107,142,94,83), (1,63,11)(2,12,64)(3,57,13)(4,14,58)(5,59,15)(6,16,60)(7,61,9)(8,10,62)(17,30,92)(18,93,31)(19,32,94)(20,95,25)(21,26,96)(22,89,27)(23,28,90)(24,91,29)(33,113,75)(34,76,114)(35,115,77)(36,78,116)(37,117,79)(38,80,118)(39,119,73)(40,74,120)(41,66,141)(42,142,67)(43,68,143)(44,144,69)(45,70,137)(46,138,71)(47,72,139)(48,140,65)(49,132,111)(50,112,133)(51,134,105)(52,106,135)(53,136,107)(54,108,129)(55,130,109)(56,110,131)(81,121,101)(82,102,122)(83,123,103)(84,104,124)(85,125,97)(86,98,126)(87,127,99)(88,100,128), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,50,91)(10,92,51)(11,52,93)(12,94,53)(13,54,95)(14,96,55)(15,56,89)(16,90,49)(17,134,62)(18,63,135)(19,136,64)(20,57,129)(21,130,58)(22,59,131)(23,132,60)(24,61,133)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,122,117)(42,118,123)(43,124,119)(44,120,125)(45,126,113)(46,114,127)(47,128,115)(48,116,121)(65,78,81)(66,82,79)(67,80,83)(68,84,73)(69,74,85)(70,86,75)(71,76,87)(72,88,77), (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)>;
G:=Group( (1,37,52,66,18,122)(2,38,53,67,19,123)(3,39,54,68,20,124)(4,40,55,69,21,125)(5,33,56,70,22,126)(6,34,49,71,23,127)(7,35,50,72,24,128)(8,36,51,65,17,121)(9,77,133,47,29,100)(10,78,134,48,30,101)(11,79,135,41,31,102)(12,80,136,42,32,103)(13,73,129,43,25,104)(14,74,130,44,26,97)(15,75,131,45,27,98)(16,76,132,46,28,99)(57,119,108,143,95,84)(58,120,109,144,96,85)(59,113,110,137,89,86)(60,114,111,138,90,87)(61,115,112,139,91,88)(62,116,105,140,92,81)(63,117,106,141,93,82)(64,118,107,142,94,83), (1,63,11)(2,12,64)(3,57,13)(4,14,58)(5,59,15)(6,16,60)(7,61,9)(8,10,62)(17,30,92)(18,93,31)(19,32,94)(20,95,25)(21,26,96)(22,89,27)(23,28,90)(24,91,29)(33,113,75)(34,76,114)(35,115,77)(36,78,116)(37,117,79)(38,80,118)(39,119,73)(40,74,120)(41,66,141)(42,142,67)(43,68,143)(44,144,69)(45,70,137)(46,138,71)(47,72,139)(48,140,65)(49,132,111)(50,112,133)(51,134,105)(52,106,135)(53,136,107)(54,108,129)(55,130,109)(56,110,131)(81,121,101)(82,102,122)(83,123,103)(84,104,124)(85,125,97)(86,98,126)(87,127,99)(88,100,128), (1,106,31)(2,32,107)(3,108,25)(4,26,109)(5,110,27)(6,28,111)(7,112,29)(8,30,105)(9,50,91)(10,92,51)(11,52,93)(12,94,53)(13,54,95)(14,96,55)(15,56,89)(16,90,49)(17,134,62)(18,63,135)(19,136,64)(20,57,129)(21,130,58)(22,59,131)(23,132,60)(24,61,133)(33,137,98)(34,99,138)(35,139,100)(36,101,140)(37,141,102)(38,103,142)(39,143,104)(40,97,144)(41,122,117)(42,118,123)(43,124,119)(44,120,125)(45,126,113)(46,114,127)(47,128,115)(48,116,121)(65,78,81)(66,82,79)(67,80,83)(68,84,73)(69,74,85)(70,86,75)(71,76,87)(72,88,77), (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) );
G=PermutationGroup([[(1,37,52,66,18,122),(2,38,53,67,19,123),(3,39,54,68,20,124),(4,40,55,69,21,125),(5,33,56,70,22,126),(6,34,49,71,23,127),(7,35,50,72,24,128),(8,36,51,65,17,121),(9,77,133,47,29,100),(10,78,134,48,30,101),(11,79,135,41,31,102),(12,80,136,42,32,103),(13,73,129,43,25,104),(14,74,130,44,26,97),(15,75,131,45,27,98),(16,76,132,46,28,99),(57,119,108,143,95,84),(58,120,109,144,96,85),(59,113,110,137,89,86),(60,114,111,138,90,87),(61,115,112,139,91,88),(62,116,105,140,92,81),(63,117,106,141,93,82),(64,118,107,142,94,83)], [(1,63,11),(2,12,64),(3,57,13),(4,14,58),(5,59,15),(6,16,60),(7,61,9),(8,10,62),(17,30,92),(18,93,31),(19,32,94),(20,95,25),(21,26,96),(22,89,27),(23,28,90),(24,91,29),(33,113,75),(34,76,114),(35,115,77),(36,78,116),(37,117,79),(38,80,118),(39,119,73),(40,74,120),(41,66,141),(42,142,67),(43,68,143),(44,144,69),(45,70,137),(46,138,71),(47,72,139),(48,140,65),(49,132,111),(50,112,133),(51,134,105),(52,106,135),(53,136,107),(54,108,129),(55,130,109),(56,110,131),(81,121,101),(82,102,122),(83,123,103),(84,104,124),(85,125,97),(86,98,126),(87,127,99),(88,100,128)], [(1,106,31),(2,32,107),(3,108,25),(4,26,109),(5,110,27),(6,28,111),(7,112,29),(8,30,105),(9,50,91),(10,92,51),(11,52,93),(12,94,53),(13,54,95),(14,96,55),(15,56,89),(16,90,49),(17,134,62),(18,63,135),(19,136,64),(20,57,129),(21,130,58),(22,59,131),(23,132,60),(24,61,133),(33,137,98),(34,99,138),(35,139,100),(36,101,140),(37,141,102),(38,103,142),(39,143,104),(40,97,144),(41,122,117),(42,118,123),(43,124,119),(44,120,125),(45,126,113),(46,114,127),(47,128,115),(48,116,121),(65,78,81),(66,82,79),(67,80,83),(68,84,73),(69,74,85),(70,86,75),(71,76,87),(72,88,77)], [(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)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6AP | 8A | ··· | 8H | 12A | ··· | 12H | 12I | ··· | 12BD | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | |||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | Dic3 | D6 | Dic3 | C3×S3 | C3⋊C8 | C3×Dic3 | S3×C6 | C3×Dic3 | C3×C3⋊C8 |
| kernel | C6×C32⋊4C8 | C3×C32⋊4C8 | C3×C6×C12 | C2×C32⋊4C8 | C32×C12 | C3×C62 | C32⋊4C8 | C6×C12 | C32×C6 | C3×C12 | C62 | C3×C6 | C6×C12 | C3×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 | 4 | 4 | 4 | 4 | 8 | 16 | 8 | 8 | 8 | 32 |
Matrix representation of C6×C32⋊4C8 ►in GL5(𝔽73)
| 64 | 0 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 72 | 0 |
| 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 34 | 50 |
| 0 | 0 | 0 | 11 | 39 |
G:=sub<GL(5,GF(73))| [64,0,0,0,0,0,64,0,0,0,0,0,64,0,0,0,0,0,72,0,0,0,0,0,72],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,72,72,0,0,0,1,0],[1,0,0,0,0,0,8,0,0,0,0,0,64,0,0,0,0,0,72,72,0,0,0,1,0],[51,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,34,11,0,0,0,50,39] >;
C6×C32⋊4C8 in GAP, Magma, Sage, TeX
C_6\times C_3^2\rtimes_4C_8
% in TeX
G:=Group("C6xC3^2:4C8"); // GroupNames label
G:=SmallGroup(432,485);
// by ID
G=gap.SmallGroup(432,485);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^-1>;
// generators/relations