direct product, metabelian, supersoluble, monomial
Aliases: C4×C12⋊S3, C12⋊5D12, C122⋊8C2, C62.217C23, C12⋊9(C4×S3), (C4×C12)⋊9S3, C3⋊3(C4×D12), (C3×C12)⋊20D4, C32⋊17(C4×D4), C42⋊5(C3⋊S3), C6.49(C2×D12), (C2×C12).356D6, C6.94(C4○D12), C12⋊Dic3⋊25C2, C6.11D12⋊28C2, (C6×C12).286C22, C2.3(C12.59D6), C4⋊2(C4×C3⋊S3), C6.64(S3×C2×C4), (C3×C12)⋊17(C2×C4), C2.1(C2×C12⋊S3), (C3×C6).189(C2×D4), (C3×C6).95(C22×C4), (C2×C12⋊S3).17C2, (C3×C6).110(C4○D4), (C2×C6).234(C22×S3), C22.11(C22×C3⋊S3), (C22×C3⋊S3).80C22, (C2×C3⋊Dic3).152C22, C2.6(C2×C4×C3⋊S3), (C2×C4×C3⋊S3)⋊16C2, (C2×C3⋊S3)⋊9(C2×C4), (C2×C4).97(C2×C3⋊S3), SmallGroup(288,730)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C12⋊S3
G = < a,b,c,d | a4=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 1124 in 282 conjugacy classes, 97 normal (21 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×C12, C22×S3, C4×D4, C3⋊Dic3, C3×C12, C3×C12, C2×C3⋊S3, C2×C3⋊S3, C62, C4⋊Dic3, D6⋊C4, C4×C12, S3×C2×C4, C2×D12, C4×C3⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C4×D12, C12⋊Dic3, C6.11D12, C122, C2×C4×C3⋊S3, C2×C12⋊S3, C4×C12⋊S3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C3⋊S3, C4×S3, D12, C22×S3, C4×D4, C2×C3⋊S3, S3×C2×C4, C2×D12, C4○D12, C4×C3⋊S3, C12⋊S3, C22×C3⋊S3, C4×D12, C2×C4×C3⋊S3, C2×C12⋊S3, C12.59D6, C4×C12⋊S3
►On 144 points
Generators in S144
(1 39 94 77)(2 40 95 78)(3 41 96 79)(4 42 85 80)(5 43 86 81)(6 44 87 82)(7 45 88 83)(8 46 89 84)(9 47 90 73)(10 48 91 74)(11 37 92 75)(12 38 93 76)(13 130 140 102)(14 131 141 103)(15 132 142 104)(16 121 143 105)(17 122 144 106)(18 123 133 107)(19 124 134 108)(20 125 135 97)(21 126 136 98)(22 127 137 99)(23 128 138 100)(24 129 139 101)(25 69 113 55)(26 70 114 56)(27 71 115 57)(28 72 116 58)(29 61 117 59)(30 62 118 60)(31 63 119 49)(32 64 120 50)(33 65 109 51)(34 66 110 52)(35 67 111 53)(36 68 112 54)
(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)
(1 126 28)(2 127 29)(3 128 30)(4 129 31)(5 130 32)(6 131 33)(7 132 34)(8 121 35)(9 122 36)(10 123 25)(11 124 26)(12 125 27)(13 50 81)(14 51 82)(15 52 83)(16 53 84)(17 54 73)(18 55 74)(19 56 75)(20 57 76)(21 58 77)(22 59 78)(23 60 79)(24 49 80)(37 134 70)(38 135 71)(39 136 72)(40 137 61)(41 138 62)(42 139 63)(43 140 64)(44 141 65)(45 142 66)(46 143 67)(47 144 68)(48 133 69)(85 101 119)(86 102 120)(87 103 109)(88 104 110)(89 105 111)(90 106 112)(91 107 113)(92 108 114)(93 97 115)(94 98 116)(95 99 117)(96 100 118)
(1 4)(2 3)(5 12)(6 11)(7 10)(8 9)(13 57)(14 56)(15 55)(16 54)(17 53)(18 52)(19 51)(20 50)(21 49)(22 60)(23 59)(24 58)(25 132)(26 131)(27 130)(28 129)(29 128)(30 127)(31 126)(32 125)(33 124)(34 123)(35 122)(36 121)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)(61 138)(62 137)(63 136)(64 135)(65 134)(66 133)(67 144)(68 143)(69 142)(70 141)(71 140)(72 139)(73 84)(74 83)(75 82)(76 81)(77 80)(78 79)(85 94)(86 93)(87 92)(88 91)(89 90)(95 96)(97 120)(98 119)(99 118)(100 117)(101 116)(102 115)(103 114)(104 113)(105 112)(106 111)(107 110)(108 109)
G:=sub<Sym(144)| (1,39,94,77)(2,40,95,78)(3,41,96,79)(4,42,85,80)(5,43,86,81)(6,44,87,82)(7,45,88,83)(8,46,89,84)(9,47,90,73)(10,48,91,74)(11,37,92,75)(12,38,93,76)(13,130,140,102)(14,131,141,103)(15,132,142,104)(16,121,143,105)(17,122,144,106)(18,123,133,107)(19,124,134,108)(20,125,135,97)(21,126,136,98)(22,127,137,99)(23,128,138,100)(24,129,139,101)(25,69,113,55)(26,70,114,56)(27,71,115,57)(28,72,116,58)(29,61,117,59)(30,62,118,60)(31,63,119,49)(32,64,120,50)(33,65,109,51)(34,66,110,52)(35,67,111,53)(36,68,112,54), (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), (1,126,28)(2,127,29)(3,128,30)(4,129,31)(5,130,32)(6,131,33)(7,132,34)(8,121,35)(9,122,36)(10,123,25)(11,124,26)(12,125,27)(13,50,81)(14,51,82)(15,52,83)(16,53,84)(17,54,73)(18,55,74)(19,56,75)(20,57,76)(21,58,77)(22,59,78)(23,60,79)(24,49,80)(37,134,70)(38,135,71)(39,136,72)(40,137,61)(41,138,62)(42,139,63)(43,140,64)(44,141,65)(45,142,66)(46,143,67)(47,144,68)(48,133,69)(85,101,119)(86,102,120)(87,103,109)(88,104,110)(89,105,111)(90,106,112)(91,107,113)(92,108,114)(93,97,115)(94,98,116)(95,99,117)(96,100,118), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,132)(26,131)(27,130)(28,129)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109)>;
G:=Group( (1,39,94,77)(2,40,95,78)(3,41,96,79)(4,42,85,80)(5,43,86,81)(6,44,87,82)(7,45,88,83)(8,46,89,84)(9,47,90,73)(10,48,91,74)(11,37,92,75)(12,38,93,76)(13,130,140,102)(14,131,141,103)(15,132,142,104)(16,121,143,105)(17,122,144,106)(18,123,133,107)(19,124,134,108)(20,125,135,97)(21,126,136,98)(22,127,137,99)(23,128,138,100)(24,129,139,101)(25,69,113,55)(26,70,114,56)(27,71,115,57)(28,72,116,58)(29,61,117,59)(30,62,118,60)(31,63,119,49)(32,64,120,50)(33,65,109,51)(34,66,110,52)(35,67,111,53)(36,68,112,54), (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), (1,126,28)(2,127,29)(3,128,30)(4,129,31)(5,130,32)(6,131,33)(7,132,34)(8,121,35)(9,122,36)(10,123,25)(11,124,26)(12,125,27)(13,50,81)(14,51,82)(15,52,83)(16,53,84)(17,54,73)(18,55,74)(19,56,75)(20,57,76)(21,58,77)(22,59,78)(23,60,79)(24,49,80)(37,134,70)(38,135,71)(39,136,72)(40,137,61)(41,138,62)(42,139,63)(43,140,64)(44,141,65)(45,142,66)(46,143,67)(47,144,68)(48,133,69)(85,101,119)(86,102,120)(87,103,109)(88,104,110)(89,105,111)(90,106,112)(91,107,113)(92,108,114)(93,97,115)(94,98,116)(95,99,117)(96,100,118), (1,4)(2,3)(5,12)(6,11)(7,10)(8,9)(13,57)(14,56)(15,55)(16,54)(17,53)(18,52)(19,51)(20,50)(21,49)(22,60)(23,59)(24,58)(25,132)(26,131)(27,130)(28,129)(29,128)(30,127)(31,126)(32,125)(33,124)(34,123)(35,122)(36,121)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)(61,138)(62,137)(63,136)(64,135)(65,134)(66,133)(67,144)(68,143)(69,142)(70,141)(71,140)(72,139)(73,84)(74,83)(75,82)(76,81)(77,80)(78,79)(85,94)(86,93)(87,92)(88,91)(89,90)(95,96)(97,120)(98,119)(99,118)(100,117)(101,116)(102,115)(103,114)(104,113)(105,112)(106,111)(107,110)(108,109) );
G=PermutationGroup([[(1,39,94,77),(2,40,95,78),(3,41,96,79),(4,42,85,80),(5,43,86,81),(6,44,87,82),(7,45,88,83),(8,46,89,84),(9,47,90,73),(10,48,91,74),(11,37,92,75),(12,38,93,76),(13,130,140,102),(14,131,141,103),(15,132,142,104),(16,121,143,105),(17,122,144,106),(18,123,133,107),(19,124,134,108),(20,125,135,97),(21,126,136,98),(22,127,137,99),(23,128,138,100),(24,129,139,101),(25,69,113,55),(26,70,114,56),(27,71,115,57),(28,72,116,58),(29,61,117,59),(30,62,118,60),(31,63,119,49),(32,64,120,50),(33,65,109,51),(34,66,110,52),(35,67,111,53),(36,68,112,54)], [(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)], [(1,126,28),(2,127,29),(3,128,30),(4,129,31),(5,130,32),(6,131,33),(7,132,34),(8,121,35),(9,122,36),(10,123,25),(11,124,26),(12,125,27),(13,50,81),(14,51,82),(15,52,83),(16,53,84),(17,54,73),(18,55,74),(19,56,75),(20,57,76),(21,58,77),(22,59,78),(23,60,79),(24,49,80),(37,134,70),(38,135,71),(39,136,72),(40,137,61),(41,138,62),(42,139,63),(43,140,64),(44,141,65),(45,142,66),(46,143,67),(47,144,68),(48,133,69),(85,101,119),(86,102,120),(87,103,109),(88,104,110),(89,105,111),(90,106,112),(91,107,113),(92,108,114),(93,97,115),(94,98,116),(95,99,117),(96,100,118)], [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9),(13,57),(14,56),(15,55),(16,54),(17,53),(18,52),(19,51),(20,50),(21,49),(22,60),(23,59),(24,58),(25,132),(26,131),(27,130),(28,129),(29,128),(30,127),(31,126),(32,125),(33,124),(34,123),(35,122),(36,121),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47),(61,138),(62,137),(63,136),(64,135),(65,134),(66,133),(67,144),(68,143),(69,142),(70,141),(71,140),(72,139),(73,84),(74,83),(75,82),(76,81),(77,80),(78,79),(85,94),(86,93),(87,92),(88,91),(89,90),(95,96),(97,120),(98,119),(99,118),(100,117),(101,116),(102,115),(103,114),(104,113),(105,112),(106,111),(107,110),(108,109)]])
84 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | ··· | 6L | 12A | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 18 | 18 | 18 | 18 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 2 | ··· | 2 |
84 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | C4○D4 | C4×S3 | D12 | C4○D12 |
| kernel | C4×C12⋊S3 | C12⋊Dic3 | C6.11D12 | C122 | C2×C4×C3⋊S3 | C2×C12⋊S3 | C12⋊S3 | C4×C12 | C3×C12 | C2×C12 | C3×C6 | C12 | C12 | C6 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 4 | 2 | 12 | 2 | 16 | 16 | 16 |
Matrix representation of C4×C12⋊S3 ►in GL4(𝔽13) generated by
| 5 | 0 | 0 | 0 |
| 0 | 5 | 0 | 0 |
| 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 5 |
| 12 | 1 | 0 | 0 |
| 12 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 |
| 0 | 0 | 6 | 4 |
| 0 | 12 | 0 | 0 |
| 1 | 12 | 0 | 0 |
| 0 | 0 | 1 | 10 |
| 0 | 0 | 1 | 11 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 9 | 8 |
| 0 | 0 | 3 | 4 |
G:=sub<GL(4,GF(13))| [5,0,0,0,0,5,0,0,0,0,5,0,0,0,0,5],[12,12,0,0,1,0,0,0,0,0,9,6,0,0,8,4],[0,1,0,0,12,12,0,0,0,0,1,1,0,0,10,11],[0,1,0,0,1,0,0,0,0,0,9,3,0,0,8,4] >;
C4×C12⋊S3 in GAP, Magma, Sage, TeX
C_4\times C_{12}\rtimes S_3 % in TeX
G:=Group("C4xC12:S3"); // GroupNames label
G:=SmallGroup(288,730);
// by ID
G=gap.SmallGroup(288,730);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^12=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations