Copied to
clipboard

G = C2×C242S3order 288 = 25·32

Direct product of C2 and C242S3

direct product, metabelian, supersoluble, monomial

Aliases: C2×C242S3, C2424D6, C12.48D12, C62.90D4, (C2×C24)⋊7S3, (C6×C24)⋊11C2, (C3×C6)⋊7SD16, C61(C24⋊C2), (C2×C6).38D12, C6.56(C2×D12), (C3×C24)⋊26C22, (C3×C12).123D4, (C2×C12).384D6, C4.6(C12⋊S3), C3214(C2×SD16), (C3×C12).151C23, C12.189(C22×S3), (C6×C12).300C22, C324Q813C22, C12⋊S3.22C22, C22.12(C12⋊S3), C88(C2×C3⋊S3), (C2×C8)⋊5(C3⋊S3), C32(C2×C24⋊C2), (C3×C6).196(C2×D4), C4.26(C22×C3⋊S3), (C2×C324Q8)⋊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

C1C3×C12 — C2×C242S3
C1C3C32C3×C6C3×C12C12⋊S3C2×C12⋊S3 — C2×C242S3
C32C3×C6C3×C12 — C2×C242S3
C1C22C2×C4C2×C8

Generators and relations for C2×C242S3
 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, C324Q8, C324Q8, C12⋊S3, C12⋊S3, C2×C3⋊Dic3, C6×C12, C22×C3⋊S3, C2×C24⋊C2, C242S3, C6×C24, C2×C324Q8, C2×C12⋊S3, C2×C242S3
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, C242S3, C2×C12⋊S3, C2×C242S3

Smallest permutation representation of C2×C242S3
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 2A2B2C2D2E3A3B3C3D4A4B4C4D6A···6L8A8B8C8D12A···12P24A···24AF
order122222333344446···6888812···1224···24
size1111363622222236362···222222···22···2

78 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2S3D4D4D6D6SD16D12D12C24⋊C2
kernelC2×C242S3C242S3C6×C24C2×C324Q8C2×C12⋊S3C2×C24C3×C12C62C24C2×C12C3×C6C12C2×C6C6
# reps141114118448832

Matrix representation of C2×C242S3 in GL4(𝔽73) generated by

72000
07200
00720
00072
,
66700
665900
001125
004836
,
0100
727200
0010
0001
,
72000
1100
0001
0010
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×C242S3 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

׿
×
𝔽