direct product, metabelian, supersoluble, monomial, A-group
Aliases: C3⋊S3×C24, C3⋊2(S3×C24), C24⋊4(C3×S3), (C3×C24)⋊15C6, (C3×C24)⋊14S3, C33⋊17(C2×C8), C6.23(S3×C12), (C32×C24)⋊9C2, C32⋊12(S3×C8), C12.101(S3×C6), C32⋊10(C2×C24), C32⋊4C8⋊17C6, (C3×C12).223D6, C3⋊Dic3.9C12, (C32×C12).91C22, C4.12(C6×C3⋊S3), C6.24(C4×C3⋊S3), C2.1(C12×C3⋊S3), (C4×C3⋊S3).13C6, (C6×C3⋊S3).14C4, (C2×C3⋊S3).9C12, C12.94(C2×C3⋊S3), (C3×C6).75(C4×S3), (C12×C3⋊S3).20C2, (C3×C12).96(C2×C6), (C3×C6).47(C2×C12), (C3×C32⋊4C8)⋊21C2, (C3×C3⋊Dic3).14C4, (C32×C6).54(C2×C4), SmallGroup(432,480)
Series: Derived ►Chief ►Lower central ►Upper central
| C32 — C3⋊S3×C24 |
Generators and relations for C3⋊S3×C24
G = < a,b,c,d | a24=b3=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b-1, dcd=c-1 >
Subgroups: 420 in 164 conjugacy classes, 62 normal (26 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, C8, C2×C4, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C3⋊C8, C24, C24, C24, C4×S3, C2×C12, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, C32×C6, C3×C3⋊C8, C32⋊4C8, C3×C24, C3×C24, C3×C24, S3×C12, C4×C3⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, S3×C24, C8×C3⋊S3, C3×C32⋊4C8, C32×C24, C12×C3⋊S3, C3⋊S3×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C2×C8, C3×S3, C3⋊S3, C24, C4×S3, C2×C12, S3×C6, C2×C3⋊S3, S3×C8, C2×C24, C3×C3⋊S3, S3×C12, C4×C3⋊S3, C6×C3⋊S3, S3×C24, C8×C3⋊S3, C12×C3⋊S3, C3⋊S3×C24
►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)
(1 131 36)(2 132 37)(3 133 38)(4 134 39)(5 135 40)(6 136 41)(7 137 42)(8 138 43)(9 139 44)(10 140 45)(11 141 46)(12 142 47)(13 143 48)(14 144 25)(15 121 26)(16 122 27)(17 123 28)(18 124 29)(19 125 30)(20 126 31)(21 127 32)(22 128 33)(23 129 34)(24 130 35)(49 104 74)(50 105 75)(51 106 76)(52 107 77)(53 108 78)(54 109 79)(55 110 80)(56 111 81)(57 112 82)(58 113 83)(59 114 84)(60 115 85)(61 116 86)(62 117 87)(63 118 88)(64 119 89)(65 120 90)(66 97 91)(67 98 92)(68 99 93)(69 100 94)(70 101 95)(71 102 96)(72 103 73)
(1 44 123)(2 45 124)(3 46 125)(4 47 126)(5 48 127)(6 25 128)(7 26 129)(8 27 130)(9 28 131)(10 29 132)(11 30 133)(12 31 134)(13 32 135)(14 33 136)(15 34 137)(16 35 138)(17 36 139)(18 37 140)(19 38 141)(20 39 142)(21 40 143)(22 41 144)(23 42 121)(24 43 122)(49 90 112)(50 91 113)(51 92 114)(52 93 115)(53 94 116)(54 95 117)(55 96 118)(56 73 119)(57 74 120)(58 75 97)(59 76 98)(60 77 99)(61 78 100)(62 79 101)(63 80 102)(64 81 103)(65 82 104)(66 83 105)(67 84 106)(68 85 107)(69 86 108)(70 87 109)(71 88 110)(72 89 111)
(1 95)(2 96)(3 73)(4 74)(5 75)(6 76)(7 77)(8 78)(9 79)(10 80)(11 81)(12 82)(13 83)(14 84)(15 85)(16 86)(17 87)(18 88)(19 89)(20 90)(21 91)(22 92)(23 93)(24 94)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 68)(35 69)(36 70)(37 71)(38 72)(39 49)(40 50)(41 51)(42 52)(43 53)(44 54)(45 55)(46 56)(47 57)(48 58)(97 127)(98 128)(99 129)(100 130)(101 131)(102 132)(103 133)(104 134)(105 135)(106 136)(107 137)(108 138)(109 139)(110 140)(111 141)(112 142)(113 143)(114 144)(115 121)(116 122)(117 123)(118 124)(119 125)(120 126)
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), (1,131,36)(2,132,37)(3,133,38)(4,134,39)(5,135,40)(6,136,41)(7,137,42)(8,138,43)(9,139,44)(10,140,45)(11,141,46)(12,142,47)(13,143,48)(14,144,25)(15,121,26)(16,122,27)(17,123,28)(18,124,29)(19,125,30)(20,126,31)(21,127,32)(22,128,33)(23,129,34)(24,130,35)(49,104,74)(50,105,75)(51,106,76)(52,107,77)(53,108,78)(54,109,79)(55,110,80)(56,111,81)(57,112,82)(58,113,83)(59,114,84)(60,115,85)(61,116,86)(62,117,87)(63,118,88)(64,119,89)(65,120,90)(66,97,91)(67,98,92)(68,99,93)(69,100,94)(70,101,95)(71,102,96)(72,103,73), (1,44,123)(2,45,124)(3,46,125)(4,47,126)(5,48,127)(6,25,128)(7,26,129)(8,27,130)(9,28,131)(10,29,132)(11,30,133)(12,31,134)(13,32,135)(14,33,136)(15,34,137)(16,35,138)(17,36,139)(18,37,140)(19,38,141)(20,39,142)(21,40,143)(22,41,144)(23,42,121)(24,43,122)(49,90,112)(50,91,113)(51,92,114)(52,93,115)(53,94,116)(54,95,117)(55,96,118)(56,73,119)(57,74,120)(58,75,97)(59,76,98)(60,77,99)(61,78,100)(62,79,101)(63,80,102)(64,81,103)(65,82,104)(66,83,105)(67,84,106)(68,85,107)(69,86,108)(70,87,109)(71,88,110)(72,89,111), (1,95)(2,96)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126)>;
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), (1,131,36)(2,132,37)(3,133,38)(4,134,39)(5,135,40)(6,136,41)(7,137,42)(8,138,43)(9,139,44)(10,140,45)(11,141,46)(12,142,47)(13,143,48)(14,144,25)(15,121,26)(16,122,27)(17,123,28)(18,124,29)(19,125,30)(20,126,31)(21,127,32)(22,128,33)(23,129,34)(24,130,35)(49,104,74)(50,105,75)(51,106,76)(52,107,77)(53,108,78)(54,109,79)(55,110,80)(56,111,81)(57,112,82)(58,113,83)(59,114,84)(60,115,85)(61,116,86)(62,117,87)(63,118,88)(64,119,89)(65,120,90)(66,97,91)(67,98,92)(68,99,93)(69,100,94)(70,101,95)(71,102,96)(72,103,73), (1,44,123)(2,45,124)(3,46,125)(4,47,126)(5,48,127)(6,25,128)(7,26,129)(8,27,130)(9,28,131)(10,29,132)(11,30,133)(12,31,134)(13,32,135)(14,33,136)(15,34,137)(16,35,138)(17,36,139)(18,37,140)(19,38,141)(20,39,142)(21,40,143)(22,41,144)(23,42,121)(24,43,122)(49,90,112)(50,91,113)(51,92,114)(52,93,115)(53,94,116)(54,95,117)(55,96,118)(56,73,119)(57,74,120)(58,75,97)(59,76,98)(60,77,99)(61,78,100)(62,79,101)(63,80,102)(64,81,103)(65,82,104)(66,83,105)(67,84,106)(68,85,107)(69,86,108)(70,87,109)(71,88,110)(72,89,111), (1,95)(2,96)(3,73)(4,74)(5,75)(6,76)(7,77)(8,78)(9,79)(10,80)(11,81)(12,82)(13,83)(14,84)(15,85)(16,86)(17,87)(18,88)(19,89)(20,90)(21,91)(22,92)(23,93)(24,94)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68)(35,69)(36,70)(37,71)(38,72)(39,49)(40,50)(41,51)(42,52)(43,53)(44,54)(45,55)(46,56)(47,57)(48,58)(97,127)(98,128)(99,129)(100,130)(101,131)(102,132)(103,133)(104,134)(105,135)(106,136)(107,137)(108,138)(109,139)(110,140)(111,141)(112,142)(113,143)(114,144)(115,121)(116,122)(117,123)(118,124)(119,125)(120,126) );
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)], [(1,131,36),(2,132,37),(3,133,38),(4,134,39),(5,135,40),(6,136,41),(7,137,42),(8,138,43),(9,139,44),(10,140,45),(11,141,46),(12,142,47),(13,143,48),(14,144,25),(15,121,26),(16,122,27),(17,123,28),(18,124,29),(19,125,30),(20,126,31),(21,127,32),(22,128,33),(23,129,34),(24,130,35),(49,104,74),(50,105,75),(51,106,76),(52,107,77),(53,108,78),(54,109,79),(55,110,80),(56,111,81),(57,112,82),(58,113,83),(59,114,84),(60,115,85),(61,116,86),(62,117,87),(63,118,88),(64,119,89),(65,120,90),(66,97,91),(67,98,92),(68,99,93),(69,100,94),(70,101,95),(71,102,96),(72,103,73)], [(1,44,123),(2,45,124),(3,46,125),(4,47,126),(5,48,127),(6,25,128),(7,26,129),(8,27,130),(9,28,131),(10,29,132),(11,30,133),(12,31,134),(13,32,135),(14,33,136),(15,34,137),(16,35,138),(17,36,139),(18,37,140),(19,38,141),(20,39,142),(21,40,143),(22,41,144),(23,42,121),(24,43,122),(49,90,112),(50,91,113),(51,92,114),(52,93,115),(53,94,116),(54,95,117),(55,96,118),(56,73,119),(57,74,120),(58,75,97),(59,76,98),(60,77,99),(61,78,100),(62,79,101),(63,80,102),(64,81,103),(65,82,104),(66,83,105),(67,84,106),(68,85,107),(69,86,108),(70,87,109),(71,88,110),(72,89,111)], [(1,95),(2,96),(3,73),(4,74),(5,75),(6,76),(7,77),(8,78),(9,79),(10,80),(11,81),(12,82),(13,83),(14,84),(15,85),(16,86),(17,87),(18,88),(19,89),(20,90),(21,91),(22,92),(23,93),(24,94),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68),(35,69),(36,70),(37,71),(38,72),(39,49),(40,50),(41,51),(42,52),(43,53),(44,54),(45,55),(46,56),(47,57),(48,58),(97,127),(98,128),(99,129),(100,130),(101,131),(102,132),(103,133),(104,134),(105,135),(106,136),(107,137),(108,138),(109,139),(110,140),(111,141),(112,142),(113,143),(114,144),(115,121),(116,122),(117,123),(118,124),(119,125),(120,126)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 4D | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 6Q | 6R | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12AB | 12AC | 12AD | 12AE | 12AF | 24A | ··· | 24H | 24I | ··· | 24BD | 24BE | ··· | 24BL |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 9 | 9 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 9 | 9 | 9 | 9 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 9 | 9 | 9 | 9 | 1 | ··· | 1 | 2 | ··· | 2 | 9 | ··· | 9 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C12 | S3×C24 |
| kernel | C3⋊S3×C24 | C3×C32⋊4C8 | C32×C24 | C12×C3⋊S3 | C8×C3⋊S3 | C3×C3⋊Dic3 | C6×C3⋊S3 | C32⋊4C8 | C3×C24 | C4×C3⋊S3 | C3×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3⋊S3 | C3×C24 | C3×C12 | C24 | C3×C6 | C12 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 4 | 4 | 8 | 8 | 8 | 16 | 16 | 32 |
Matrix representation of C3⋊S3×C24 ►in GL4(𝔽73) generated by
| 43 | 0 | 0 | 0 |
| 0 | 43 | 0 | 0 |
| 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 65 |
| 64 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 65 | 8 |
| 0 | 72 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 7 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [43,0,0,0,0,43,0,0,0,0,65,0,0,0,0,65],[64,0,0,0,0,8,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,64,65,0,0,0,8],[0,72,0,0,72,0,0,0,0,0,1,0,0,0,7,72] >;
C3⋊S3×C24 in GAP, Magma, Sage, TeX
C_3\rtimes S_3\times C_{24} % in TeX
G:=Group("C3:S3xC24"); // GroupNames label
G:=SmallGroup(432,480);
// by ID
G=gap.SmallGroup(432,480);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,92,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^24=b^3=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