direct product, metabelian, supersoluble, monomial
Aliases: C3×C24⋊2S3, C33⋊20SD16, C24⋊2(C3×S3), C24⋊6(C3⋊S3), (C3×C24)⋊11C6, (C3×C24)⋊12S3, C12.80(S3×C6), (C32×C24)⋊6C2, (C3×C6).61D12, C6.19(C3×D12), C32⋊4Q8⋊7C6, C12⋊S3.3C6, (C3×C12).207D6, (C32×C6).57D4, C32⋊11(C3×SD16), C6.22(C12⋊S3), C32⋊10(C24⋊C2), (C32×C12).82C22, C8⋊2(C3×C3⋊S3), C4.8(C6×C3⋊S3), C3⋊1(C3×C24⋊C2), C12.84(C2×C3⋊S3), (C3×C6).49(C3×D4), C2.3(C3×C12⋊S3), (C3×C12).72(C2×C6), (C3×C12⋊S3).6C2, (C3×C32⋊4Q8)⋊16C2, SmallGroup(432,482)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊2S3
G = < a,b,c,d | a3=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b11, dcd=c-1 >
Subgroups: 596 in 152 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C3, C3, C4, C4, C22, S3, C6, C6, C6, C8, D4, Q8, C32, C32, C32, Dic3, C12, C12, C12, D6, C2×C6, SD16, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, C24, C24, C24, Dic6, D12, C3×D4, C3×Q8, C33, C3×Dic3, C3⋊Dic3, C3×C12, C3×C12, C3×C12, S3×C6, C2×C3⋊S3, C24⋊C2, C3×SD16, C3×C3⋊S3, C32×C6, C3×C24, C3×C24, C3×C24, C3×Dic6, C3×D12, C32⋊4Q8, C12⋊S3, C3×C3⋊Dic3, C32×C12, C6×C3⋊S3, C3×C24⋊C2, C24⋊2S3, C32×C24, C3×C32⋊4Q8, C3×C12⋊S3, C3×C24⋊2S3
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, SD16, C3×S3, C3⋊S3, D12, C3×D4, S3×C6, C2×C3⋊S3, C24⋊C2, C3×SD16, C3×C3⋊S3, C3×D12, C12⋊S3, C6×C3⋊S3, C3×C24⋊C2, C24⋊2S3, C3×C12⋊S3, C3×C24⋊2S3
►On 144 points
Generators in S144
(1 115 45)(2 116 46)(3 117 47)(4 118 48)(5 119 25)(6 120 26)(7 97 27)(8 98 28)(9 99 29)(10 100 30)(11 101 31)(12 102 32)(13 103 33)(14 104 34)(15 105 35)(16 106 36)(17 107 37)(18 108 38)(19 109 39)(20 110 40)(21 111 41)(22 112 42)(23 113 43)(24 114 44)(49 92 121)(50 93 122)(51 94 123)(52 95 124)(53 96 125)(54 73 126)(55 74 127)(56 75 128)(57 76 129)(58 77 130)(59 78 131)(60 79 132)(61 80 133)(62 81 134)(63 82 135)(64 83 136)(65 84 137)(66 85 138)(67 86 139)(68 87 140)(69 88 141)(70 89 142)(71 90 143)(72 91 144)
(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)
(1 107 29)(2 108 30)(3 109 31)(4 110 32)(5 111 33)(6 112 34)(7 113 35)(8 114 36)(9 115 37)(10 116 38)(11 117 39)(12 118 40)(13 119 41)(14 120 42)(15 97 43)(16 98 44)(17 99 45)(18 100 46)(19 101 47)(20 102 48)(21 103 25)(22 104 26)(23 105 27)(24 106 28)(49 137 76)(50 138 77)(51 139 78)(52 140 79)(53 141 80)(54 142 81)(55 143 82)(56 144 83)(57 121 84)(58 122 85)(59 123 86)(60 124 87)(61 125 88)(62 126 89)(63 127 90)(64 128 91)(65 129 92)(66 130 93)(67 131 94)(68 132 95)(69 133 96)(70 134 73)(71 135 74)(72 136 75)
(1 82)(2 93)(3 80)(4 91)(5 78)(6 89)(7 76)(8 87)(9 74)(10 85)(11 96)(12 83)(13 94)(14 81)(15 92)(16 79)(17 90)(18 77)(19 88)(20 75)(21 86)(22 73)(23 84)(24 95)(25 59)(26 70)(27 57)(28 68)(29 55)(30 66)(31 53)(32 64)(33 51)(34 62)(35 49)(36 60)(37 71)(38 58)(39 69)(40 56)(41 67)(42 54)(43 65)(44 52)(45 63)(46 50)(47 61)(48 72)(97 129)(98 140)(99 127)(100 138)(101 125)(102 136)(103 123)(104 134)(105 121)(106 132)(107 143)(108 130)(109 141)(110 128)(111 139)(112 126)(113 137)(114 124)(115 135)(116 122)(117 133)(118 144)(119 131)(120 142)
G:=sub<Sym(144)| (1,115,45)(2,116,46)(3,117,47)(4,118,48)(5,119,25)(6,120,26)(7,97,27)(8,98,28)(9,99,29)(10,100,30)(11,101,31)(12,102,32)(13,103,33)(14,104,34)(15,105,35)(16,106,36)(17,107,37)(18,108,38)(19,109,39)(20,110,40)(21,111,41)(22,112,42)(23,113,43)(24,114,44)(49,92,121)(50,93,122)(51,94,123)(52,95,124)(53,96,125)(54,73,126)(55,74,127)(56,75,128)(57,76,129)(58,77,130)(59,78,131)(60,79,132)(61,80,133)(62,81,134)(63,82,135)(64,83,136)(65,84,137)(66,85,138)(67,86,139)(68,87,140)(69,88,141)(70,89,142)(71,90,143)(72,91,144), (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), (1,107,29)(2,108,30)(3,109,31)(4,110,32)(5,111,33)(6,112,34)(7,113,35)(8,114,36)(9,115,37)(10,116,38)(11,117,39)(12,118,40)(13,119,41)(14,120,42)(15,97,43)(16,98,44)(17,99,45)(18,100,46)(19,101,47)(20,102,48)(21,103,25)(22,104,26)(23,105,27)(24,106,28)(49,137,76)(50,138,77)(51,139,78)(52,140,79)(53,141,80)(54,142,81)(55,143,82)(56,144,83)(57,121,84)(58,122,85)(59,123,86)(60,124,87)(61,125,88)(62,126,89)(63,127,90)(64,128,91)(65,129,92)(66,130,93)(67,131,94)(68,132,95)(69,133,96)(70,134,73)(71,135,74)(72,136,75), (1,82)(2,93)(3,80)(4,91)(5,78)(6,89)(7,76)(8,87)(9,74)(10,85)(11,96)(12,83)(13,94)(14,81)(15,92)(16,79)(17,90)(18,77)(19,88)(20,75)(21,86)(22,73)(23,84)(24,95)(25,59)(26,70)(27,57)(28,68)(29,55)(30,66)(31,53)(32,64)(33,51)(34,62)(35,49)(36,60)(37,71)(38,58)(39,69)(40,56)(41,67)(42,54)(43,65)(44,52)(45,63)(46,50)(47,61)(48,72)(97,129)(98,140)(99,127)(100,138)(101,125)(102,136)(103,123)(104,134)(105,121)(106,132)(107,143)(108,130)(109,141)(110,128)(111,139)(112,126)(113,137)(114,124)(115,135)(116,122)(117,133)(118,144)(119,131)(120,142)>;
G:=Group( (1,115,45)(2,116,46)(3,117,47)(4,118,48)(5,119,25)(6,120,26)(7,97,27)(8,98,28)(9,99,29)(10,100,30)(11,101,31)(12,102,32)(13,103,33)(14,104,34)(15,105,35)(16,106,36)(17,107,37)(18,108,38)(19,109,39)(20,110,40)(21,111,41)(22,112,42)(23,113,43)(24,114,44)(49,92,121)(50,93,122)(51,94,123)(52,95,124)(53,96,125)(54,73,126)(55,74,127)(56,75,128)(57,76,129)(58,77,130)(59,78,131)(60,79,132)(61,80,133)(62,81,134)(63,82,135)(64,83,136)(65,84,137)(66,85,138)(67,86,139)(68,87,140)(69,88,141)(70,89,142)(71,90,143)(72,91,144), (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), (1,107,29)(2,108,30)(3,109,31)(4,110,32)(5,111,33)(6,112,34)(7,113,35)(8,114,36)(9,115,37)(10,116,38)(11,117,39)(12,118,40)(13,119,41)(14,120,42)(15,97,43)(16,98,44)(17,99,45)(18,100,46)(19,101,47)(20,102,48)(21,103,25)(22,104,26)(23,105,27)(24,106,28)(49,137,76)(50,138,77)(51,139,78)(52,140,79)(53,141,80)(54,142,81)(55,143,82)(56,144,83)(57,121,84)(58,122,85)(59,123,86)(60,124,87)(61,125,88)(62,126,89)(63,127,90)(64,128,91)(65,129,92)(66,130,93)(67,131,94)(68,132,95)(69,133,96)(70,134,73)(71,135,74)(72,136,75), (1,82)(2,93)(3,80)(4,91)(5,78)(6,89)(7,76)(8,87)(9,74)(10,85)(11,96)(12,83)(13,94)(14,81)(15,92)(16,79)(17,90)(18,77)(19,88)(20,75)(21,86)(22,73)(23,84)(24,95)(25,59)(26,70)(27,57)(28,68)(29,55)(30,66)(31,53)(32,64)(33,51)(34,62)(35,49)(36,60)(37,71)(38,58)(39,69)(40,56)(41,67)(42,54)(43,65)(44,52)(45,63)(46,50)(47,61)(48,72)(97,129)(98,140)(99,127)(100,138)(101,125)(102,136)(103,123)(104,134)(105,121)(106,132)(107,143)(108,130)(109,141)(110,128)(111,139)(112,126)(113,137)(114,124)(115,135)(116,122)(117,133)(118,144)(119,131)(120,142) );
G=PermutationGroup([[(1,115,45),(2,116,46),(3,117,47),(4,118,48),(5,119,25),(6,120,26),(7,97,27),(8,98,28),(9,99,29),(10,100,30),(11,101,31),(12,102,32),(13,103,33),(14,104,34),(15,105,35),(16,106,36),(17,107,37),(18,108,38),(19,109,39),(20,110,40),(21,111,41),(22,112,42),(23,113,43),(24,114,44),(49,92,121),(50,93,122),(51,94,123),(52,95,124),(53,96,125),(54,73,126),(55,74,127),(56,75,128),(57,76,129),(58,77,130),(59,78,131),(60,79,132),(61,80,133),(62,81,134),(63,82,135),(64,83,136),(65,84,137),(66,85,138),(67,86,139),(68,87,140),(69,88,141),(70,89,142),(71,90,143),(72,91,144)], [(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)], [(1,107,29),(2,108,30),(3,109,31),(4,110,32),(5,111,33),(6,112,34),(7,113,35),(8,114,36),(9,115,37),(10,116,38),(11,117,39),(12,118,40),(13,119,41),(14,120,42),(15,97,43),(16,98,44),(17,99,45),(18,100,46),(19,101,47),(20,102,48),(21,103,25),(22,104,26),(23,105,27),(24,106,28),(49,137,76),(50,138,77),(51,139,78),(52,140,79),(53,141,80),(54,142,81),(55,143,82),(56,144,83),(57,121,84),(58,122,85),(59,123,86),(60,124,87),(61,125,88),(62,126,89),(63,127,90),(64,128,91),(65,129,92),(66,130,93),(67,131,94),(68,132,95),(69,133,96),(70,134,73),(71,135,74),(72,136,75)], [(1,82),(2,93),(3,80),(4,91),(5,78),(6,89),(7,76),(8,87),(9,74),(10,85),(11,96),(12,83),(13,94),(14,81),(15,92),(16,79),(17,90),(18,77),(19,88),(20,75),(21,86),(22,73),(23,84),(24,95),(25,59),(26,70),(27,57),(28,68),(29,55),(30,66),(31,53),(32,64),(33,51),(34,62),(35,49),(36,60),(37,71),(38,58),(39,69),(40,56),(41,67),(42,54),(43,65),(44,52),(45,63),(46,50),(47,61),(48,72),(97,129),(98,140),(99,127),(100,138),(101,125),(102,136),(103,123),(104,134),(105,121),(106,132),(107,143),(108,130),(109,141),(110,128),(111,139),(112,126),(113,137),(114,124),(115,135),(116,122),(117,133),(118,144),(119,131),(120,142)]])
117 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | ··· | 3N | 4A | 4B | 6A | 6B | 6C | ··· | 6N | 6O | 6P | 8A | 8B | 12A | ··· | 12Z | 12AA | 12AB | 24A | ··· | 24AZ |
| order | 1 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 36 | 1 | 1 | 2 | ··· | 2 | 2 | 36 | 1 | 1 | 2 | ··· | 2 | 36 | 36 | 2 | 2 | 2 | ··· | 2 | 36 | 36 | 2 | ··· | 2 |
117 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | SD16 | C3×S3 | D12 | C3×D4 | S3×C6 | C24⋊C2 | C3×SD16 | C3×D12 | C3×C24⋊C2 |
| kernel | C3×C24⋊2S3 | C32×C24 | C3×C32⋊4Q8 | C3×C12⋊S3 | C24⋊2S3 | C3×C24 | C32⋊4Q8 | C12⋊S3 | C3×C24 | C32×C6 | C3×C12 | C33 | C24 | C3×C6 | C3×C6 | C12 | C32 | C32 | C6 | C3 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 1 | 4 | 2 | 8 | 8 | 2 | 8 | 16 | 4 | 16 | 32 |
Matrix representation of C3×C24⋊2S3 ►in GL4(𝔽73) generated by
| 8 | 0 | 0 | 0 |
| 0 | 8 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 56 | 0 |
| 0 | 0 | 0 | 43 |
| 8 | 0 | 0 | 0 |
| 0 | 64 | 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],[1,0,0,0,0,1,0,0,0,0,56,0,0,0,0,43],[8,0,0,0,0,64,0,0,0,0,8,0,0,0,0,64],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C24⋊2S3 in GAP, Magma, Sage, TeX
C_3\times C_{24}\rtimes_2S_3 % in TeX
G:=Group("C3xC24:2S3"); // GroupNames label
G:=SmallGroup(432,482);
// by ID
G=gap.SmallGroup(432,482);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-3,-3,197,92,1011,80,4037,14118]);
// 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^11,d*c*d=c^-1>;
// generators/relations