direct product, metabelian, supersoluble, monomial, A-group
Aliases: C4×C3⋊Dic3, C12⋊2Dic3, C32⋊4C42, C62.21C22, (C3×C12)⋊6C4, C6.13(C4×S3), C3⋊2(C4×Dic3), (C2×C6).30D6, (C6×C12).13C2, (C2×C12).17S3, C6.14(C2×Dic3), C2.2(C4×C3⋊S3), (C2×C4).6(C3⋊S3), (C3×C6).24(C2×C4), C22.3(C2×C3⋊S3), C2.2(C2×C3⋊Dic3), (C2×C3⋊Dic3).10C2, SmallGroup(144,92)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C4×C3⋊Dic3 |
Generators and relations for C4×C3⋊Dic3
G = < a,b,c,d | a4=b3=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c-1 >
Subgroups: 202 in 90 conjugacy classes, 55 normal (9 characteristic)
C1, C2, C2, C3, C4, C4, C22, C6, C2×C4, C2×C4, C32, Dic3, C12, C2×C6, C42, C3×C6, C3×C6, C2×Dic3, C2×C12, C3⋊Dic3, C3×C12, C62, C4×Dic3, C2×C3⋊Dic3, C6×C12, C4×C3⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, C42, C3⋊S3, C4×S3, C2×Dic3, C3⋊Dic3, C2×C3⋊S3, C4×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C4×C3⋊Dic3
►Regular action on 144 points
Generators in S144
(1 69 33 50)(2 70 34 51)(3 71 35 52)(4 72 36 53)(5 67 31 54)(6 68 32 49)(7 131 27 134)(8 132 28 135)(9 127 29 136)(10 128 30 137)(11 129 25 138)(12 130 26 133)(13 95 57 66)(14 96 58 61)(15 91 59 62)(16 92 60 63)(17 93 55 64)(18 94 56 65)(19 110 116 80)(20 111 117 81)(21 112 118 82)(22 113 119 83)(23 114 120 84)(24 109 115 79)(37 97 43 75)(38 98 44 76)(39 99 45 77)(40 100 46 78)(41 101 47 73)(42 102 48 74)(85 144 108 121)(86 139 103 122)(87 140 104 123)(88 141 105 124)(89 142 106 125)(90 143 107 126)
(1 41 60)(2 42 55)(3 37 56)(4 38 57)(5 39 58)(6 40 59)(7 125 21)(8 126 22)(9 121 23)(10 122 24)(11 123 19)(12 124 20)(13 36 44)(14 31 45)(15 32 46)(16 33 47)(17 34 48)(18 35 43)(25 140 116)(26 141 117)(27 142 118)(28 143 119)(29 144 120)(30 139 115)(49 78 91)(50 73 92)(51 74 93)(52 75 94)(53 76 95)(54 77 96)(61 67 99)(62 68 100)(63 69 101)(64 70 102)(65 71 97)(66 72 98)(79 137 103)(80 138 104)(81 133 105)(82 134 106)(83 135 107)(84 136 108)(85 114 127)(86 109 128)(87 110 129)(88 111 130)(89 112 131)(90 113 132)
(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 85 4 88)(2 90 5 87)(3 89 6 86)(7 78 10 75)(8 77 11 74)(9 76 12 73)(13 81 16 84)(14 80 17 83)(15 79 18 82)(19 93 22 96)(20 92 23 95)(21 91 24 94)(25 102 28 99)(26 101 29 98)(27 100 30 97)(31 104 34 107)(32 103 35 106)(33 108 36 105)(37 131 40 128)(38 130 41 127)(39 129 42 132)(43 134 46 137)(44 133 47 136)(45 138 48 135)(49 122 52 125)(50 121 53 124)(51 126 54 123)(55 113 58 110)(56 112 59 109)(57 111 60 114)(61 116 64 119)(62 115 65 118)(63 120 66 117)(67 140 70 143)(68 139 71 142)(69 144 72 141)
G:=sub<Sym(144)| (1,69,33,50)(2,70,34,51)(3,71,35,52)(4,72,36,53)(5,67,31,54)(6,68,32,49)(7,131,27,134)(8,132,28,135)(9,127,29,136)(10,128,30,137)(11,129,25,138)(12,130,26,133)(13,95,57,66)(14,96,58,61)(15,91,59,62)(16,92,60,63)(17,93,55,64)(18,94,56,65)(19,110,116,80)(20,111,117,81)(21,112,118,82)(22,113,119,83)(23,114,120,84)(24,109,115,79)(37,97,43,75)(38,98,44,76)(39,99,45,77)(40,100,46,78)(41,101,47,73)(42,102,48,74)(85,144,108,121)(86,139,103,122)(87,140,104,123)(88,141,105,124)(89,142,106,125)(90,143,107,126), (1,41,60)(2,42,55)(3,37,56)(4,38,57)(5,39,58)(6,40,59)(7,125,21)(8,126,22)(9,121,23)(10,122,24)(11,123,19)(12,124,20)(13,36,44)(14,31,45)(15,32,46)(16,33,47)(17,34,48)(18,35,43)(25,140,116)(26,141,117)(27,142,118)(28,143,119)(29,144,120)(30,139,115)(49,78,91)(50,73,92)(51,74,93)(52,75,94)(53,76,95)(54,77,96)(61,67,99)(62,68,100)(63,69,101)(64,70,102)(65,71,97)(66,72,98)(79,137,103)(80,138,104)(81,133,105)(82,134,106)(83,135,107)(84,136,108)(85,114,127)(86,109,128)(87,110,129)(88,111,130)(89,112,131)(90,113,132), (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), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,81,16,84)(14,80,17,83)(15,79,18,82)(19,93,22,96)(20,92,23,95)(21,91,24,94)(25,102,28,99)(26,101,29,98)(27,100,30,97)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141)>;
G:=Group( (1,69,33,50)(2,70,34,51)(3,71,35,52)(4,72,36,53)(5,67,31,54)(6,68,32,49)(7,131,27,134)(8,132,28,135)(9,127,29,136)(10,128,30,137)(11,129,25,138)(12,130,26,133)(13,95,57,66)(14,96,58,61)(15,91,59,62)(16,92,60,63)(17,93,55,64)(18,94,56,65)(19,110,116,80)(20,111,117,81)(21,112,118,82)(22,113,119,83)(23,114,120,84)(24,109,115,79)(37,97,43,75)(38,98,44,76)(39,99,45,77)(40,100,46,78)(41,101,47,73)(42,102,48,74)(85,144,108,121)(86,139,103,122)(87,140,104,123)(88,141,105,124)(89,142,106,125)(90,143,107,126), (1,41,60)(2,42,55)(3,37,56)(4,38,57)(5,39,58)(6,40,59)(7,125,21)(8,126,22)(9,121,23)(10,122,24)(11,123,19)(12,124,20)(13,36,44)(14,31,45)(15,32,46)(16,33,47)(17,34,48)(18,35,43)(25,140,116)(26,141,117)(27,142,118)(28,143,119)(29,144,120)(30,139,115)(49,78,91)(50,73,92)(51,74,93)(52,75,94)(53,76,95)(54,77,96)(61,67,99)(62,68,100)(63,69,101)(64,70,102)(65,71,97)(66,72,98)(79,137,103)(80,138,104)(81,133,105)(82,134,106)(83,135,107)(84,136,108)(85,114,127)(86,109,128)(87,110,129)(88,111,130)(89,112,131)(90,113,132), (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), (1,85,4,88)(2,90,5,87)(3,89,6,86)(7,78,10,75)(8,77,11,74)(9,76,12,73)(13,81,16,84)(14,80,17,83)(15,79,18,82)(19,93,22,96)(20,92,23,95)(21,91,24,94)(25,102,28,99)(26,101,29,98)(27,100,30,97)(31,104,34,107)(32,103,35,106)(33,108,36,105)(37,131,40,128)(38,130,41,127)(39,129,42,132)(43,134,46,137)(44,133,47,136)(45,138,48,135)(49,122,52,125)(50,121,53,124)(51,126,54,123)(55,113,58,110)(56,112,59,109)(57,111,60,114)(61,116,64,119)(62,115,65,118)(63,120,66,117)(67,140,70,143)(68,139,71,142)(69,144,72,141) );
G=PermutationGroup([[(1,69,33,50),(2,70,34,51),(3,71,35,52),(4,72,36,53),(5,67,31,54),(6,68,32,49),(7,131,27,134),(8,132,28,135),(9,127,29,136),(10,128,30,137),(11,129,25,138),(12,130,26,133),(13,95,57,66),(14,96,58,61),(15,91,59,62),(16,92,60,63),(17,93,55,64),(18,94,56,65),(19,110,116,80),(20,111,117,81),(21,112,118,82),(22,113,119,83),(23,114,120,84),(24,109,115,79),(37,97,43,75),(38,98,44,76),(39,99,45,77),(40,100,46,78),(41,101,47,73),(42,102,48,74),(85,144,108,121),(86,139,103,122),(87,140,104,123),(88,141,105,124),(89,142,106,125),(90,143,107,126)], [(1,41,60),(2,42,55),(3,37,56),(4,38,57),(5,39,58),(6,40,59),(7,125,21),(8,126,22),(9,121,23),(10,122,24),(11,123,19),(12,124,20),(13,36,44),(14,31,45),(15,32,46),(16,33,47),(17,34,48),(18,35,43),(25,140,116),(26,141,117),(27,142,118),(28,143,119),(29,144,120),(30,139,115),(49,78,91),(50,73,92),(51,74,93),(52,75,94),(53,76,95),(54,77,96),(61,67,99),(62,68,100),(63,69,101),(64,70,102),(65,71,97),(66,72,98),(79,137,103),(80,138,104),(81,133,105),(82,134,106),(83,135,107),(84,136,108),(85,114,127),(86,109,128),(87,110,129),(88,111,130),(89,112,131),(90,113,132)], [(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)], [(1,85,4,88),(2,90,5,87),(3,89,6,86),(7,78,10,75),(8,77,11,74),(9,76,12,73),(13,81,16,84),(14,80,17,83),(15,79,18,82),(19,93,22,96),(20,92,23,95),(21,91,24,94),(25,102,28,99),(26,101,29,98),(27,100,30,97),(31,104,34,107),(32,103,35,106),(33,108,36,105),(37,131,40,128),(38,130,41,127),(39,129,42,132),(43,134,46,137),(44,133,47,136),(45,138,48,135),(49,122,52,125),(50,121,53,124),(51,126,54,123),(55,113,58,110),(56,112,59,109),(57,111,60,114),(61,116,64,119),(62,115,65,118),(63,120,66,117),(67,140,70,143),(68,139,71,142),(69,144,72,141)]])
C4×C3⋊Dic3 is a maximal subgroup of
C6.(S3×C8) C2.Dic32 D12⋊4Dic3 C12.15Dic6 C12.30Dic6 C24⋊Dic3 C62.37D4 C62.39D4 (C3×C12)⋊4C8 C32⋊2C8⋊C4 C32⋊5(C4⋊C8) C62.8C23 C62.13C23 C62.25C23 C62.32C23 C62.33C23 C62.40C23 C62.42C23 C62.43C23 C4×S3×Dic3 C62.48C23 D12⋊Dic3 C62.72C23 C62.84C23 C62.85C23 C12⋊Dic6 C42×C3⋊S3 C122⋊16C2 C62.221C23 C62.223C23 C62.225C23 C62.229C23 C62.231C23 C12⋊2Dic6 C62.233C23 C62.234C23 C62.236C23 C62.237C23 C62.242C23 C62.247C23 C62.254C23 C62.258C23 C62.259C23 C62.262C23
C4×C3⋊Dic3 is a maximal quotient of
C122.C2 C24⋊Dic3 C62.15Q8
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | ··· | 4L | 6A | ··· | 6L | 12A | ··· | 12P |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 9 | ··· | 9 | 2 | ··· | 2 | 2 | ··· | 2 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | C4×S3 |
| kernel | C4×C3⋊Dic3 | C2×C3⋊Dic3 | C6×C12 | C3⋊Dic3 | C3×C12 | C2×C12 | C12 | C2×C6 | C6 |
| # reps | 1 | 2 | 1 | 8 | 4 | 4 | 8 | 4 | 16 |
Matrix representation of C4×C3⋊Dic3 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 1 | 12 |
| 0 | 0 | 1 | 0 |
| 9 | 2 | 0 | 0 |
| 11 | 4 | 0 | 0 |
| 0 | 0 | 4 | 11 |
| 0 | 0 | 2 | 9 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,12,0,0,0,0,12],[12,12,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[0,12,0,0,1,1,0,0,0,0,1,1,0,0,12,0],[9,11,0,0,2,4,0,0,0,0,4,2,0,0,11,9] >;
C4×C3⋊Dic3 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes {\rm Dic}_3 % in TeX
G:=Group("C4xC3:Dic3"); // GroupNames label
G:=SmallGroup(144,92);
// by ID
G=gap.SmallGroup(144,92);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-3,24,55,964,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^6=1,d^2=c^3,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