direct product, metabelian, supersoluble, monomial
Aliases: C3×C24⋊S3, C33⋊14M4(2), C24⋊5(C3×S3), C24⋊7(C3⋊S3), (C3×C24)⋊16C6, (C3×C24)⋊17S3, C6.24(S3×C12), C12.102(S3×C6), (C32×C24)⋊13C2, C32⋊4C8⋊15C6, (C3×C12).224D6, C3⋊Dic3.6C12, C32⋊9(C3×M4(2)), C32⋊11(C8⋊S3), (C32×C12).92C22, C8⋊3(C3×C3⋊S3), C3⋊2(C3×C8⋊S3), (C4×C3⋊S3).8C6, (C6×C3⋊S3).9C4, C2.3(C12×C3⋊S3), C6.25(C4×C3⋊S3), C4.13(C6×C3⋊S3), (C2×C3⋊S3).6C12, C12.95(C2×C3⋊S3), (C3×C6).76(C4×S3), (C12×C3⋊S3).17C2, (C3×C12).97(C2×C6), (C3×C6).48(C2×C12), (C3×C32⋊4C8)⋊19C2, (C3×C3⋊Dic3).12C4, (C32×C6).55(C2×C4), SmallGroup(432,481)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊S3
G = < a,b,c,d | a3=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >
Subgroups: 420 in 152 conjugacy classes, 58 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, M4(2), 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, C8⋊S3, C3×M4(2), 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, C3×C8⋊S3, C24⋊S3, C3×C32⋊4C8, C32×C24, C12×C3⋊S3, C3×C24⋊S3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C12, D6, C2×C6, M4(2), C3×S3, C3⋊S3, C4×S3, C2×C12, S3×C6, C2×C3⋊S3, C8⋊S3, C3×M4(2), C3×C3⋊S3, S3×C12, C4×C3⋊S3, C6×C3⋊S3, C3×C8⋊S3, C24⋊S3, C12×C3⋊S3, C3×C24⋊S3
►On 144 points
Generators in S144
(1 54 38)(2 55 39)(3 56 40)(4 57 41)(5 58 42)(6 59 43)(7 60 44)(8 61 45)(9 62 46)(10 63 47)(11 64 48)(12 65 25)(13 66 26)(14 67 27)(15 68 28)(16 69 29)(17 70 30)(18 71 31)(19 72 32)(20 49 33)(21 50 34)(22 51 35)(23 52 36)(24 53 37)(73 121 99)(74 122 100)(75 123 101)(76 124 102)(77 125 103)(78 126 104)(79 127 105)(80 128 106)(81 129 107)(82 130 108)(83 131 109)(84 132 110)(85 133 111)(86 134 112)(87 135 113)(88 136 114)(89 137 115)(90 138 116)(91 139 117)(92 140 118)(93 141 119)(94 142 120)(95 143 97)(96 144 98)
(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 30 62)(2 31 63)(3 32 64)(4 33 65)(5 34 66)(6 35 67)(7 36 68)(8 37 69)(9 38 70)(10 39 71)(11 40 72)(12 41 49)(13 42 50)(14 43 51)(15 44 52)(16 45 53)(17 46 54)(18 47 55)(19 48 56)(20 25 57)(21 26 58)(22 27 59)(23 28 60)(24 29 61)(73 137 107)(74 138 108)(75 139 109)(76 140 110)(77 141 111)(78 142 112)(79 143 113)(80 144 114)(81 121 115)(82 122 116)(83 123 117)(84 124 118)(85 125 119)(86 126 120)(87 127 97)(88 128 98)(89 129 99)(90 130 100)(91 131 101)(92 132 102)(93 133 103)(94 134 104)(95 135 105)(96 136 106)
(1 112)(2 117)(3 98)(4 103)(5 108)(6 113)(7 118)(8 99)(9 104)(10 109)(11 114)(12 119)(13 100)(14 105)(15 110)(16 115)(17 120)(18 101)(19 106)(20 111)(21 116)(22 97)(23 102)(24 107)(25 141)(26 122)(27 127)(28 132)(29 137)(30 142)(31 123)(32 128)(33 133)(34 138)(35 143)(36 124)(37 129)(38 134)(39 139)(40 144)(41 125)(42 130)(43 135)(44 140)(45 121)(46 126)(47 131)(48 136)(49 85)(50 90)(51 95)(52 76)(53 81)(54 86)(55 91)(56 96)(57 77)(58 82)(59 87)(60 92)(61 73)(62 78)(63 83)(64 88)(65 93)(66 74)(67 79)(68 84)(69 89)(70 94)(71 75)(72 80)
G:=sub<Sym(144)| (1,54,38)(2,55,39)(3,56,40)(4,57,41)(5,58,42)(6,59,43)(7,60,44)(8,61,45)(9,62,46)(10,63,47)(11,64,48)(12,65,25)(13,66,26)(14,67,27)(15,68,28)(16,69,29)(17,70,30)(18,71,31)(19,72,32)(20,49,33)(21,50,34)(22,51,35)(23,52,36)(24,53,37)(73,121,99)(74,122,100)(75,123,101)(76,124,102)(77,125,103)(78,126,104)(79,127,105)(80,128,106)(81,129,107)(82,130,108)(83,131,109)(84,132,110)(85,133,111)(86,134,112)(87,135,113)(88,136,114)(89,137,115)(90,138,116)(91,139,117)(92,140,118)(93,141,119)(94,142,120)(95,143,97)(96,144,98), (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), (1,30,62)(2,31,63)(3,32,64)(4,33,65)(5,34,66)(6,35,67)(7,36,68)(8,37,69)(9,38,70)(10,39,71)(11,40,72)(12,41,49)(13,42,50)(14,43,51)(15,44,52)(16,45,53)(17,46,54)(18,47,55)(19,48,56)(20,25,57)(21,26,58)(22,27,59)(23,28,60)(24,29,61)(73,137,107)(74,138,108)(75,139,109)(76,140,110)(77,141,111)(78,142,112)(79,143,113)(80,144,114)(81,121,115)(82,122,116)(83,123,117)(84,124,118)(85,125,119)(86,126,120)(87,127,97)(88,128,98)(89,129,99)(90,130,100)(91,131,101)(92,132,102)(93,133,103)(94,134,104)(95,135,105)(96,136,106), (1,112)(2,117)(3,98)(4,103)(5,108)(6,113)(7,118)(8,99)(9,104)(10,109)(11,114)(12,119)(13,100)(14,105)(15,110)(16,115)(17,120)(18,101)(19,106)(20,111)(21,116)(22,97)(23,102)(24,107)(25,141)(26,122)(27,127)(28,132)(29,137)(30,142)(31,123)(32,128)(33,133)(34,138)(35,143)(36,124)(37,129)(38,134)(39,139)(40,144)(41,125)(42,130)(43,135)(44,140)(45,121)(46,126)(47,131)(48,136)(49,85)(50,90)(51,95)(52,76)(53,81)(54,86)(55,91)(56,96)(57,77)(58,82)(59,87)(60,92)(61,73)(62,78)(63,83)(64,88)(65,93)(66,74)(67,79)(68,84)(69,89)(70,94)(71,75)(72,80)>;
G:=Group( (1,54,38)(2,55,39)(3,56,40)(4,57,41)(5,58,42)(6,59,43)(7,60,44)(8,61,45)(9,62,46)(10,63,47)(11,64,48)(12,65,25)(13,66,26)(14,67,27)(15,68,28)(16,69,29)(17,70,30)(18,71,31)(19,72,32)(20,49,33)(21,50,34)(22,51,35)(23,52,36)(24,53,37)(73,121,99)(74,122,100)(75,123,101)(76,124,102)(77,125,103)(78,126,104)(79,127,105)(80,128,106)(81,129,107)(82,130,108)(83,131,109)(84,132,110)(85,133,111)(86,134,112)(87,135,113)(88,136,114)(89,137,115)(90,138,116)(91,139,117)(92,140,118)(93,141,119)(94,142,120)(95,143,97)(96,144,98), (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), (1,30,62)(2,31,63)(3,32,64)(4,33,65)(5,34,66)(6,35,67)(7,36,68)(8,37,69)(9,38,70)(10,39,71)(11,40,72)(12,41,49)(13,42,50)(14,43,51)(15,44,52)(16,45,53)(17,46,54)(18,47,55)(19,48,56)(20,25,57)(21,26,58)(22,27,59)(23,28,60)(24,29,61)(73,137,107)(74,138,108)(75,139,109)(76,140,110)(77,141,111)(78,142,112)(79,143,113)(80,144,114)(81,121,115)(82,122,116)(83,123,117)(84,124,118)(85,125,119)(86,126,120)(87,127,97)(88,128,98)(89,129,99)(90,130,100)(91,131,101)(92,132,102)(93,133,103)(94,134,104)(95,135,105)(96,136,106), (1,112)(2,117)(3,98)(4,103)(5,108)(6,113)(7,118)(8,99)(9,104)(10,109)(11,114)(12,119)(13,100)(14,105)(15,110)(16,115)(17,120)(18,101)(19,106)(20,111)(21,116)(22,97)(23,102)(24,107)(25,141)(26,122)(27,127)(28,132)(29,137)(30,142)(31,123)(32,128)(33,133)(34,138)(35,143)(36,124)(37,129)(38,134)(39,139)(40,144)(41,125)(42,130)(43,135)(44,140)(45,121)(46,126)(47,131)(48,136)(49,85)(50,90)(51,95)(52,76)(53,81)(54,86)(55,91)(56,96)(57,77)(58,82)(59,87)(60,92)(61,73)(62,78)(63,83)(64,88)(65,93)(66,74)(67,79)(68,84)(69,89)(70,94)(71,75)(72,80) );
G=PermutationGroup([[(1,54,38),(2,55,39),(3,56,40),(4,57,41),(5,58,42),(6,59,43),(7,60,44),(8,61,45),(9,62,46),(10,63,47),(11,64,48),(12,65,25),(13,66,26),(14,67,27),(15,68,28),(16,69,29),(17,70,30),(18,71,31),(19,72,32),(20,49,33),(21,50,34),(22,51,35),(23,52,36),(24,53,37),(73,121,99),(74,122,100),(75,123,101),(76,124,102),(77,125,103),(78,126,104),(79,127,105),(80,128,106),(81,129,107),(82,130,108),(83,131,109),(84,132,110),(85,133,111),(86,134,112),(87,135,113),(88,136,114),(89,137,115),(90,138,116),(91,139,117),(92,140,118),(93,141,119),(94,142,120),(95,143,97),(96,144,98)], [(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)], [(1,30,62),(2,31,63),(3,32,64),(4,33,65),(5,34,66),(6,35,67),(7,36,68),(8,37,69),(9,38,70),(10,39,71),(11,40,72),(12,41,49),(13,42,50),(14,43,51),(15,44,52),(16,45,53),(17,46,54),(18,47,55),(19,48,56),(20,25,57),(21,26,58),(22,27,59),(23,28,60),(24,29,61),(73,137,107),(74,138,108),(75,139,109),(76,140,110),(77,141,111),(78,142,112),(79,143,113),(80,144,114),(81,121,115),(82,122,116),(83,123,117),(84,124,118),(85,125,119),(86,126,120),(87,127,97),(88,128,98),(89,129,99),(90,130,100),(91,131,101),(92,132,102),(93,133,103),(94,134,104),(95,135,105),(96,136,106)], [(1,112),(2,117),(3,98),(4,103),(5,108),(6,113),(7,118),(8,99),(9,104),(10,109),(11,114),(12,119),(13,100),(14,105),(15,110),(16,115),(17,120),(18,101),(19,106),(20,111),(21,116),(22,97),(23,102),(24,107),(25,141),(26,122),(27,127),(28,132),(29,137),(30,142),(31,123),(32,128),(33,133),(34,138),(35,143),(36,124),(37,129),(38,134),(39,139),(40,144),(41,125),(42,130),(43,135),(44,140),(45,121),(46,126),(47,131),(48,136),(49,85),(50,90),(51,95),(52,76),(53,81),(54,86),(55,91),(56,96),(57,77),(58,82),(59,87),(60,92),(61,73),(62,78),(63,83),(64,88),(65,93),(66,74),(67,79),(68,84),(69,89),(70,94),(71,75),(72,80)]])
126 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12AB | 12AC | 12AD | 24A | ··· | 24AZ | 24BA | 24BB | 24BC | 24BD |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 18 | 1 | 1 | 2 | ··· | 2 | 1 | 1 | 18 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 2 | 2 | 18 | 18 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 18 | 18 | 2 | ··· | 2 | 18 | 18 | 18 | 18 |
126 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 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | S3 | D6 | M4(2) | C3×S3 | C4×S3 | S3×C6 | C8⋊S3 | C3×M4(2) | S3×C12 | C3×C8⋊S3 |
| kernel | C3×C24⋊S3 | C3×C32⋊4C8 | C32×C24 | C12×C3⋊S3 | C24⋊S3 | C3×C3⋊Dic3 | C6×C3⋊S3 | C32⋊4C8 | C3×C24 | C4×C3⋊S3 | C3⋊Dic3 | C2×C3⋊S3 | C3×C24 | C3×C12 | C33 | C24 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 8 | 8 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C3×C24⋊S3 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 43 | 0 | 0 | 0 |
| 0 | 21 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 8 | 0 |
| 0 | 0 | 0 | 64 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [8,0,0,0,0,8,0,0,0,0,64,0,0,0,0,64],[43,0,0,0,0,21,0,0,0,0,72,0,0,0,0,72],[1,0,0,0,0,1,0,0,0,0,8,0,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C24⋊S3 in GAP, Magma, Sage, TeX
C_3\times C_{24}\rtimes S_3 % in TeX
G:=Group("C3xC24:S3"); // GroupNames label
G:=SmallGroup(432,481);
// by ID
G=gap.SmallGroup(432,481);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,365,92,80,4037,14118]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^24=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^5,d*c*d=c^-1>;
// generators/relations