direct product, metabelian, supersoluble, monomial
Aliases: C4×C32⋊C12, He3⋊3C42, C62.18D6, C32⋊(C4×C12), (C3×C12)⋊2C12, (C6×C12).9C6, (C4×He3)⋊5C4, C6.14(S3×C12), (C6×C12).17S3, C3⋊Dic3⋊2C12, (C3×C12)⋊3Dic3, C62.4(C2×C6), C6.12(C6×Dic3), C3.2(Dic3×C12), C32⋊2(C4×Dic3), C12.14(C3×Dic3), (C22×He3).16C22, (C4×C3⋊Dic3)⋊C3, (C3×C6).7(C2×C12), (C2×C6).38(S3×C6), (C3×C6).14(C4×S3), C2.2(C4×C32⋊C6), (C2×C4×He3).11C2, (C2×C12).32(C3×S3), C2.2(C2×C32⋊C12), (C2×C3⋊Dic3).4C6, (C3×C6).8(C2×Dic3), (C2×C32⋊C12).8C2, (C2×C4).6(C32⋊C6), (C2×He3).28(C2×C4), C22.3(C2×C32⋊C6), SmallGroup(432,138)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4×C32⋊C12 |
Generators and relations for C4×C32⋊C12
G = < a,b,c,d | a4=b3=c3=d12=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1c, dcd-1=c-1 >
Subgroups: 397 in 121 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C2×C4, C2×C4, C32, C32, Dic3, C12, C12, C2×C6, C2×C6, C42, C3×C6, C3×C6, C3×C6, C2×Dic3, C2×C12, C2×C12, He3, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C62, C62, C4×Dic3, C4×C12, C2×He3, C2×He3, C6×Dic3, C2×C3⋊Dic3, C6×C12, C6×C12, C32⋊C12, C4×He3, C22×He3, Dic3×C12, C4×C3⋊Dic3, C2×C32⋊C12, C2×C4×He3, C4×C32⋊C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, Dic3, C12, D6, C2×C6, C42, C3×S3, C4×S3, C2×Dic3, C2×C12, C3×Dic3, S3×C6, C4×Dic3, C4×C12, C32⋊C6, S3×C12, C6×Dic3, C32⋊C12, C2×C32⋊C6, Dic3×C12, C4×C32⋊C6, C2×C32⋊C12, C4×C32⋊C12
►On 144 points
Generators in S144
(1 62 120 15)(2 63 109 16)(3 64 110 17)(4 65 111 18)(5 66 112 19)(6 67 113 20)(7 68 114 21)(8 69 115 22)(9 70 116 23)(10 71 117 24)(11 72 118 13)(12 61 119 14)(25 93 41 130)(26 94 42 131)(27 95 43 132)(28 96 44 121)(29 85 45 122)(30 86 46 123)(31 87 47 124)(32 88 48 125)(33 89 37 126)(34 90 38 127)(35 91 39 128)(36 92 40 129)(49 137 106 78)(50 138 107 79)(51 139 108 80)(52 140 97 81)(53 141 98 82)(54 142 99 83)(55 143 100 84)(56 144 101 73)(57 133 102 74)(58 134 103 75)(59 135 104 76)(60 136 105 77)
(1 73 85)(2 74 86)(4 88 76)(5 89 77)(7 79 91)(8 80 92)(10 94 82)(11 95 83)(13 27 99)(15 101 29)(16 102 30)(18 32 104)(19 33 105)(21 107 35)(22 108 36)(24 26 98)(37 60 66)(39 68 50)(40 69 51)(42 53 71)(43 54 72)(45 62 56)(46 63 57)(48 59 65)(109 133 123)(111 125 135)(112 126 136)(114 138 128)(115 139 129)(117 131 141)(118 132 142)(120 144 122)
(1 85 73)(2 74 86)(3 87 75)(4 76 88)(5 89 77)(6 78 90)(7 91 79)(8 80 92)(9 93 81)(10 82 94)(11 95 83)(12 84 96)(13 27 99)(14 100 28)(15 29 101)(16 102 30)(17 31 103)(18 104 32)(19 33 105)(20 106 34)(21 35 107)(22 108 36)(23 25 97)(24 98 26)(37 60 66)(38 67 49)(39 50 68)(40 69 51)(41 52 70)(42 71 53)(43 54 72)(44 61 55)(45 56 62)(46 63 57)(47 58 64)(48 65 59)(109 133 123)(110 124 134)(111 135 125)(112 126 136)(113 137 127)(114 128 138)(115 139 129)(116 130 140)(117 141 131)(118 132 142)(119 143 121)(120 122 144)
(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)
G:=sub<Sym(144)| (1,62,120,15)(2,63,109,16)(3,64,110,17)(4,65,111,18)(5,66,112,19)(6,67,113,20)(7,68,114,21)(8,69,115,22)(9,70,116,23)(10,71,117,24)(11,72,118,13)(12,61,119,14)(25,93,41,130)(26,94,42,131)(27,95,43,132)(28,96,44,121)(29,85,45,122)(30,86,46,123)(31,87,47,124)(32,88,48,125)(33,89,37,126)(34,90,38,127)(35,91,39,128)(36,92,40,129)(49,137,106,78)(50,138,107,79)(51,139,108,80)(52,140,97,81)(53,141,98,82)(54,142,99,83)(55,143,100,84)(56,144,101,73)(57,133,102,74)(58,134,103,75)(59,135,104,76)(60,136,105,77), (1,73,85)(2,74,86)(4,88,76)(5,89,77)(7,79,91)(8,80,92)(10,94,82)(11,95,83)(13,27,99)(15,101,29)(16,102,30)(18,32,104)(19,33,105)(21,107,35)(22,108,36)(24,26,98)(37,60,66)(39,68,50)(40,69,51)(42,53,71)(43,54,72)(45,62,56)(46,63,57)(48,59,65)(109,133,123)(111,125,135)(112,126,136)(114,138,128)(115,139,129)(117,131,141)(118,132,142)(120,144,122), (1,85,73)(2,74,86)(3,87,75)(4,76,88)(5,89,77)(6,78,90)(7,91,79)(8,80,92)(9,93,81)(10,82,94)(11,95,83)(12,84,96)(13,27,99)(14,100,28)(15,29,101)(16,102,30)(17,31,103)(18,104,32)(19,33,105)(20,106,34)(21,35,107)(22,108,36)(23,25,97)(24,98,26)(37,60,66)(38,67,49)(39,50,68)(40,69,51)(41,52,70)(42,71,53)(43,54,72)(44,61,55)(45,56,62)(46,63,57)(47,58,64)(48,65,59)(109,133,123)(110,124,134)(111,135,125)(112,126,136)(113,137,127)(114,128,138)(115,139,129)(116,130,140)(117,141,131)(118,132,142)(119,143,121)(120,122,144), (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)>;
G:=Group( (1,62,120,15)(2,63,109,16)(3,64,110,17)(4,65,111,18)(5,66,112,19)(6,67,113,20)(7,68,114,21)(8,69,115,22)(9,70,116,23)(10,71,117,24)(11,72,118,13)(12,61,119,14)(25,93,41,130)(26,94,42,131)(27,95,43,132)(28,96,44,121)(29,85,45,122)(30,86,46,123)(31,87,47,124)(32,88,48,125)(33,89,37,126)(34,90,38,127)(35,91,39,128)(36,92,40,129)(49,137,106,78)(50,138,107,79)(51,139,108,80)(52,140,97,81)(53,141,98,82)(54,142,99,83)(55,143,100,84)(56,144,101,73)(57,133,102,74)(58,134,103,75)(59,135,104,76)(60,136,105,77), (1,73,85)(2,74,86)(4,88,76)(5,89,77)(7,79,91)(8,80,92)(10,94,82)(11,95,83)(13,27,99)(15,101,29)(16,102,30)(18,32,104)(19,33,105)(21,107,35)(22,108,36)(24,26,98)(37,60,66)(39,68,50)(40,69,51)(42,53,71)(43,54,72)(45,62,56)(46,63,57)(48,59,65)(109,133,123)(111,125,135)(112,126,136)(114,138,128)(115,139,129)(117,131,141)(118,132,142)(120,144,122), (1,85,73)(2,74,86)(3,87,75)(4,76,88)(5,89,77)(6,78,90)(7,91,79)(8,80,92)(9,93,81)(10,82,94)(11,95,83)(12,84,96)(13,27,99)(14,100,28)(15,29,101)(16,102,30)(17,31,103)(18,104,32)(19,33,105)(20,106,34)(21,35,107)(22,108,36)(23,25,97)(24,98,26)(37,60,66)(38,67,49)(39,50,68)(40,69,51)(41,52,70)(42,71,53)(43,54,72)(44,61,55)(45,56,62)(46,63,57)(47,58,64)(48,65,59)(109,133,123)(110,124,134)(111,135,125)(112,126,136)(113,137,127)(114,128,138)(115,139,129)(116,130,140)(117,141,131)(118,132,142)(119,143,121)(120,122,144), (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) );
G=PermutationGroup([[(1,62,120,15),(2,63,109,16),(3,64,110,17),(4,65,111,18),(5,66,112,19),(6,67,113,20),(7,68,114,21),(8,69,115,22),(9,70,116,23),(10,71,117,24),(11,72,118,13),(12,61,119,14),(25,93,41,130),(26,94,42,131),(27,95,43,132),(28,96,44,121),(29,85,45,122),(30,86,46,123),(31,87,47,124),(32,88,48,125),(33,89,37,126),(34,90,38,127),(35,91,39,128),(36,92,40,129),(49,137,106,78),(50,138,107,79),(51,139,108,80),(52,140,97,81),(53,141,98,82),(54,142,99,83),(55,143,100,84),(56,144,101,73),(57,133,102,74),(58,134,103,75),(59,135,104,76),(60,136,105,77)], [(1,73,85),(2,74,86),(4,88,76),(5,89,77),(7,79,91),(8,80,92),(10,94,82),(11,95,83),(13,27,99),(15,101,29),(16,102,30),(18,32,104),(19,33,105),(21,107,35),(22,108,36),(24,26,98),(37,60,66),(39,68,50),(40,69,51),(42,53,71),(43,54,72),(45,62,56),(46,63,57),(48,59,65),(109,133,123),(111,125,135),(112,126,136),(114,138,128),(115,139,129),(117,131,141),(118,132,142),(120,144,122)], [(1,85,73),(2,74,86),(3,87,75),(4,76,88),(5,89,77),(6,78,90),(7,91,79),(8,80,92),(9,93,81),(10,82,94),(11,95,83),(12,84,96),(13,27,99),(14,100,28),(15,29,101),(16,102,30),(17,31,103),(18,104,32),(19,33,105),(20,106,34),(21,35,107),(22,108,36),(23,25,97),(24,98,26),(37,60,66),(38,67,49),(39,50,68),(40,69,51),(41,52,70),(42,71,53),(43,54,72),(44,61,55),(45,56,62),(46,63,57),(47,58,64),(48,65,59),(109,133,123),(110,124,134),(111,135,125),(112,126,136),(113,137,127),(114,128,138),(115,139,129),(116,130,140),(117,141,131),(118,132,142),(119,143,121),(120,122,144)], [(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)]])
80 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 3F | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | 6B | 6C | 6D | ··· | 6I | 6J | ··· | 6R | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 12M | ··· | 12X | 12Y | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 3 | 3 | 6 | 6 | 6 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | ··· | 6 | 9 | ··· | 9 |
80 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 |
| type | + | + | + | + | - | + | + | - | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C12 | C12 | S3 | Dic3 | D6 | C3×S3 | C4×S3 | C3×Dic3 | S3×C6 | S3×C12 | C32⋊C6 | C32⋊C12 | C2×C32⋊C6 | C4×C32⋊C6 |
| kernel | C4×C32⋊C12 | C2×C32⋊C12 | C2×C4×He3 | C4×C3⋊Dic3 | C32⋊C12 | C4×He3 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C3×C12 | C6×C12 | C3×C12 | C62 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 | C2×C4 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 8 | 4 | 4 | 2 | 16 | 8 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 2 | 1 | 4 |
Matrix representation of C4×C32⋊C12 ►in GL8(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5 |
| 12 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 11 | 0 | 4 | 0 | 12 |
| 0 | 0 | 0 | 9 | 0 | 1 | 0 | 4 |
| 0 | 0 | 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 9 | 2 | 0 | 9 | 0 | 1 |
| 0 | 0 | 0 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 9 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 3 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 9 | 10 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3 |
| 0 | 5 | 0 | 0 | 0 | 0 | 0 | 0 |
| 5 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 6 | 9 |
| 0 | 0 | 11 | 8 | 7 | 0 | 8 | 0 |
| 0 | 0 | 6 | 7 | 0 | 12 | 0 | 10 |
| 0 | 0 | 12 | 4 | 0 | 5 | 0 | 2 |
| 0 | 0 | 0 | 0 | 6 | 9 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 7 | 0 | 0 |
G:=sub<GL(8,GF(13))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,5],[12,12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,3,0,0,0,0,0,0,0,11,9,0,0,0,0,0,0,0,0,9,0,0,0,0,0,4,1,10,3,0,0,0,0,0,0,0,0,1,0,0,0,12,4,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,9,0,0,0,0,0,0,0,2,3,0,0,0,0,0,0,0,0,9,0,0,0,0,0,9,0,10,3,0,0,0,0,0,0,0,0,9,0,0,0,1,0,0,0,10,3],[0,5,0,0,0,0,0,0,5,0,0,0,0,0,0,0,0,0,0,11,6,12,0,0,0,0,0,8,7,4,0,0,0,0,0,7,0,0,6,12,0,0,0,0,12,5,9,7,0,0,6,8,0,0,0,0,0,0,9,0,10,2,0,0] >;
C4×C32⋊C12 in GAP, Magma, Sage, TeX
C_4\times C_3^2\rtimes C_{12} % in TeX
G:=Group("C4xC3^2:C12"); // GroupNames label
G:=SmallGroup(432,138);
// by ID
G=gap.SmallGroup(432,138);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,84,176,4037,2035,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=d^12=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1*c,d*c*d^-1=c^-1>;
// generators/relations