direct product, metabelian, supersoluble, monomial
Aliases: C2×C24⋊2S3, C24⋊24D6, C12.48D12, C62.90D4, (C2×C24)⋊7S3, (C6×C24)⋊11C2, (C3×C6)⋊7SD16, C6⋊1(C24⋊C2), (C2×C6).38D12, C6.56(C2×D12), (C3×C24)⋊26C22, (C3×C12).123D4, (C2×C12).384D6, C4.6(C12⋊S3), C32⋊14(C2×SD16), (C3×C12).151C23, C12.189(C22×S3), (C6×C12).300C22, C32⋊4Q8⋊13C22, C12⋊S3.22C22, C22.12(C12⋊S3), C8⋊8(C2×C3⋊S3), (C2×C8)⋊5(C3⋊S3), C3⋊2(C2×C24⋊C2), (C3×C6).196(C2×D4), C4.26(C22×C3⋊S3), (C2×C32⋊4Q8)⋊7C2, (C2×C12⋊S3).6C2, C2.11(C2×C12⋊S3), (C2×C4).79(C2×C3⋊S3), SmallGroup(288,759)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×C24⋊2S3
G = < a,b,c,d | a2=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b11, dcd=c-1 >
Subgroups: 1012 in 204 conjugacy classes, 77 normal (17 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, D6, C2×C6, C2×C8, SD16, C2×D4, C2×Q8, C3⋊S3, C3×C6, C3×C6, C24, Dic6, D12, C2×Dic3, C2×C12, C22×S3, C2×SD16, C3⋊Dic3, C3×C12, C2×C3⋊S3, C62, C24⋊C2, C2×C24, C2×Dic6, C2×D12, C3×C24, C32⋊4Q8, C32⋊4Q8, C12⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C24⋊C2, C24⋊2S3, C6×C24, C2×C32⋊4Q8, C2×C12⋊S3, C2×C24⋊2S3
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊S3, D12, C22×S3, C2×SD16, C2×C3⋊S3, C24⋊C2, C2×D12, C12⋊S3, C22×C3⋊S3, C2×C24⋊C2, C24⋊2S3, C2×C12⋊S3, C2×C24⋊2S3
►On 144 points
Generators in S144
(1 66)(2 67)(3 68)(4 69)(5 70)(6 71)(7 72)(8 49)(9 50)(10 51)(11 52)(12 53)(13 54)(14 55)(15 56)(16 57)(17 58)(18 59)(19 60)(20 61)(21 62)(22 63)(23 64)(24 65)(25 122)(26 123)(27 124)(28 125)(29 126)(30 127)(31 128)(32 129)(33 130)(34 131)(35 132)(36 133)(37 134)(38 135)(39 136)(40 137)(41 138)(42 139)(43 140)(44 141)(45 142)(46 143)(47 144)(48 121)(73 120)(74 97)(75 98)(76 99)(77 100)(78 101)(79 102)(80 103)(81 104)(82 105)(83 106)(84 107)(85 108)(86 109)(87 110)(88 111)(89 112)(90 113)(91 114)(92 115)(93 116)(94 117)(95 118)(96 119)
(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 40 92)(2 41 93)(3 42 94)(4 43 95)(5 44 96)(6 45 73)(7 46 74)(8 47 75)(9 48 76)(10 25 77)(11 26 78)(12 27 79)(13 28 80)(14 29 81)(15 30 82)(16 31 83)(17 32 84)(18 33 85)(19 34 86)(20 35 87)(21 36 88)(22 37 89)(23 38 90)(24 39 91)(49 144 98)(50 121 99)(51 122 100)(52 123 101)(53 124 102)(54 125 103)(55 126 104)(56 127 105)(57 128 106)(58 129 107)(59 130 108)(60 131 109)(61 132 110)(62 133 111)(63 134 112)(64 135 113)(65 136 114)(66 137 115)(67 138 116)(68 139 117)(69 140 118)(70 141 119)(71 142 120)(72 143 97)
(2 12)(3 23)(4 10)(5 21)(6 8)(7 19)(9 17)(11 15)(14 24)(16 22)(18 20)(25 95)(26 82)(27 93)(28 80)(29 91)(30 78)(31 89)(32 76)(33 87)(34 74)(35 85)(36 96)(37 83)(38 94)(39 81)(40 92)(41 79)(42 90)(43 77)(44 88)(45 75)(46 86)(47 73)(48 84)(49 71)(50 58)(51 69)(52 56)(53 67)(55 65)(57 63)(59 61)(60 72)(62 70)(64 68)(97 131)(98 142)(99 129)(100 140)(101 127)(102 138)(103 125)(104 136)(105 123)(106 134)(107 121)(108 132)(109 143)(110 130)(111 141)(112 128)(113 139)(114 126)(115 137)(116 124)(117 135)(118 122)(119 133)(120 144)
G:=sub<Sym(144)| (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,121)(73,120)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119), (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), (1,40,92)(2,41,93)(3,42,94)(4,43,95)(5,44,96)(6,45,73)(7,46,74)(8,47,75)(9,48,76)(10,25,77)(11,26,78)(12,27,79)(13,28,80)(14,29,81)(15,30,82)(16,31,83)(17,32,84)(18,33,85)(19,34,86)(20,35,87)(21,36,88)(22,37,89)(23,38,90)(24,39,91)(49,144,98)(50,121,99)(51,122,100)(52,123,101)(53,124,102)(54,125,103)(55,126,104)(56,127,105)(57,128,106)(58,129,107)(59,130,108)(60,131,109)(61,132,110)(62,133,111)(63,134,112)(64,135,113)(65,136,114)(66,137,115)(67,138,116)(68,139,117)(69,140,118)(70,141,119)(71,142,120)(72,143,97), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,95)(26,82)(27,93)(28,80)(29,91)(30,78)(31,89)(32,76)(33,87)(34,74)(35,85)(36,96)(37,83)(38,94)(39,81)(40,92)(41,79)(42,90)(43,77)(44,88)(45,75)(46,86)(47,73)(48,84)(49,71)(50,58)(51,69)(52,56)(53,67)(55,65)(57,63)(59,61)(60,72)(62,70)(64,68)(97,131)(98,142)(99,129)(100,140)(101,127)(102,138)(103,125)(104,136)(105,123)(106,134)(107,121)(108,132)(109,143)(110,130)(111,141)(112,128)(113,139)(114,126)(115,137)(116,124)(117,135)(118,122)(119,133)(120,144)>;
G:=Group( (1,66)(2,67)(3,68)(4,69)(5,70)(6,71)(7,72)(8,49)(9,50)(10,51)(11,52)(12,53)(13,54)(14,55)(15,56)(16,57)(17,58)(18,59)(19,60)(20,61)(21,62)(22,63)(23,64)(24,65)(25,122)(26,123)(27,124)(28,125)(29,126)(30,127)(31,128)(32,129)(33,130)(34,131)(35,132)(36,133)(37,134)(38,135)(39,136)(40,137)(41,138)(42,139)(43,140)(44,141)(45,142)(46,143)(47,144)(48,121)(73,120)(74,97)(75,98)(76,99)(77,100)(78,101)(79,102)(80,103)(81,104)(82,105)(83,106)(84,107)(85,108)(86,109)(87,110)(88,111)(89,112)(90,113)(91,114)(92,115)(93,116)(94,117)(95,118)(96,119), (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), (1,40,92)(2,41,93)(3,42,94)(4,43,95)(5,44,96)(6,45,73)(7,46,74)(8,47,75)(9,48,76)(10,25,77)(11,26,78)(12,27,79)(13,28,80)(14,29,81)(15,30,82)(16,31,83)(17,32,84)(18,33,85)(19,34,86)(20,35,87)(21,36,88)(22,37,89)(23,38,90)(24,39,91)(49,144,98)(50,121,99)(51,122,100)(52,123,101)(53,124,102)(54,125,103)(55,126,104)(56,127,105)(57,128,106)(58,129,107)(59,130,108)(60,131,109)(61,132,110)(62,133,111)(63,134,112)(64,135,113)(65,136,114)(66,137,115)(67,138,116)(68,139,117)(69,140,118)(70,141,119)(71,142,120)(72,143,97), (2,12)(3,23)(4,10)(5,21)(6,8)(7,19)(9,17)(11,15)(14,24)(16,22)(18,20)(25,95)(26,82)(27,93)(28,80)(29,91)(30,78)(31,89)(32,76)(33,87)(34,74)(35,85)(36,96)(37,83)(38,94)(39,81)(40,92)(41,79)(42,90)(43,77)(44,88)(45,75)(46,86)(47,73)(48,84)(49,71)(50,58)(51,69)(52,56)(53,67)(55,65)(57,63)(59,61)(60,72)(62,70)(64,68)(97,131)(98,142)(99,129)(100,140)(101,127)(102,138)(103,125)(104,136)(105,123)(106,134)(107,121)(108,132)(109,143)(110,130)(111,141)(112,128)(113,139)(114,126)(115,137)(116,124)(117,135)(118,122)(119,133)(120,144) );
G=PermutationGroup([[(1,66),(2,67),(3,68),(4,69),(5,70),(6,71),(7,72),(8,49),(9,50),(10,51),(11,52),(12,53),(13,54),(14,55),(15,56),(16,57),(17,58),(18,59),(19,60),(20,61),(21,62),(22,63),(23,64),(24,65),(25,122),(26,123),(27,124),(28,125),(29,126),(30,127),(31,128),(32,129),(33,130),(34,131),(35,132),(36,133),(37,134),(38,135),(39,136),(40,137),(41,138),(42,139),(43,140),(44,141),(45,142),(46,143),(47,144),(48,121),(73,120),(74,97),(75,98),(76,99),(77,100),(78,101),(79,102),(80,103),(81,104),(82,105),(83,106),(84,107),(85,108),(86,109),(87,110),(88,111),(89,112),(90,113),(91,114),(92,115),(93,116),(94,117),(95,118),(96,119)], [(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)], [(1,40,92),(2,41,93),(3,42,94),(4,43,95),(5,44,96),(6,45,73),(7,46,74),(8,47,75),(9,48,76),(10,25,77),(11,26,78),(12,27,79),(13,28,80),(14,29,81),(15,30,82),(16,31,83),(17,32,84),(18,33,85),(19,34,86),(20,35,87),(21,36,88),(22,37,89),(23,38,90),(24,39,91),(49,144,98),(50,121,99),(51,122,100),(52,123,101),(53,124,102),(54,125,103),(55,126,104),(56,127,105),(57,128,106),(58,129,107),(59,130,108),(60,131,109),(61,132,110),(62,133,111),(63,134,112),(64,135,113),(65,136,114),(66,137,115),(67,138,116),(68,139,117),(69,140,118),(70,141,119),(71,142,120),(72,143,97)], [(2,12),(3,23),(4,10),(5,21),(6,8),(7,19),(9,17),(11,15),(14,24),(16,22),(18,20),(25,95),(26,82),(27,93),(28,80),(29,91),(30,78),(31,89),(32,76),(33,87),(34,74),(35,85),(36,96),(37,83),(38,94),(39,81),(40,92),(41,79),(42,90),(43,77),(44,88),(45,75),(46,86),(47,73),(48,84),(49,71),(50,58),(51,69),(52,56),(53,67),(55,65),(57,63),(59,61),(60,72),(62,70),(64,68),(97,131),(98,142),(99,129),(100,140),(101,127),(102,138),(103,125),(104,136),(105,123),(106,134),(107,121),(108,132),(109,143),(110,130),(111,141),(112,128),(113,139),(114,126),(115,137),(116,124),(117,135),(118,122),(119,133),(120,144)]])
78 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 4A | 4B | 4C | 4D | 6A | ··· | 6L | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 36 | 36 | 2 | 2 | 2 | 2 | 2 | 2 | 36 | 36 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
78 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||
| image | C1 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | SD16 | D12 | D12 | C24⋊C2 |
| kernel | C2×C24⋊2S3 | C24⋊2S3 | C6×C24 | C2×C32⋊4Q8 | C2×C12⋊S3 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C12 | C2×C6 | C6 |
| # reps | 1 | 4 | 1 | 1 | 1 | 4 | 1 | 1 | 8 | 4 | 4 | 8 | 8 | 32 |
Matrix representation of C2×C24⋊2S3 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 66 | 7 | 0 | 0 |
| 66 | 59 | 0 | 0 |
| 0 | 0 | 11 | 25 |
| 0 | 0 | 48 | 36 |
| 0 | 1 | 0 | 0 |
| 72 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,72,0,0,0,0,72],[66,66,0,0,7,59,0,0,0,0,11,48,0,0,25,36],[0,72,0,0,1,72,0,0,0,0,1,0,0,0,0,1],[72,1,0,0,0,1,0,0,0,0,0,1,0,0,1,0] >;
C2×C24⋊2S3 in GAP, Magma, Sage, TeX
C_2\times C_{24}\rtimes_2S_3 % in TeX
G:=Group("C2xC24:2S3"); // GroupNames label
G:=SmallGroup(288,759);
// by ID
G=gap.SmallGroup(288,759);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,254,58,675,80,2693,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=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