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G = S3×C52C8order 240 = 24·3·5

Direct product of S3 and C52C8

direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary

Aliases: S3×C52C8, C20.30D6, C12.30D10, D6.2Dic5, C60.30C22, Dic3.2Dic5, C55(S3×C8), C156(C2×C8), (C5×S3)⋊2C8, (C4×S3).3D5, C153C810C2, C4.23(S3×D5), (S3×C20).2C2, (S3×C10).3C4, C10.17(C4×S3), C30.23(C2×C4), C2.1(S3×Dic5), C6.1(C2×Dic5), (C5×Dic3).3C4, C31(C2×C52C8), (C3×C52C8)⋊4C2, SmallGroup(240,8)

Series: Derived Chief Lower central Upper central

C1C15 — S3×C52C8
C1C5C15C30C60C3×C52C8 — S3×C52C8
C15 — S3×C52C8
C1C4

Generators and relations for S3×C52C8
 G = < a,b,c,d | a3=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

3C2
3C2
3C22
3C4
3C10
3C10
3C2×C4
5C8
15C8
3C20
3C2×C10
15C2×C8
5C3⋊C8
5C24
3C2×C20
3C52C8
5S3×C8
3C2×C52C8

Smallest permutation representation of S3×C52C8
On 120 points
Generators in S120
(1 11 50)(2 12 51)(3 13 52)(4 14 53)(5 15 54)(6 16 55)(7 9 56)(8 10 49)(17 106 29)(18 107 30)(19 108 31)(20 109 32)(21 110 25)(22 111 26)(23 112 27)(24 105 28)(33 79 113)(34 80 114)(35 73 115)(36 74 116)(37 75 117)(38 76 118)(39 77 119)(40 78 120)(41 58 103)(42 59 104)(43 60 97)(44 61 98)(45 62 99)(46 63 100)(47 64 101)(48 57 102)(65 95 87)(66 96 88)(67 89 81)(68 90 82)(69 91 83)(70 92 84)(71 93 85)(72 94 86)
(1 5)(2 6)(3 7)(4 8)(9 52)(10 53)(11 54)(12 55)(13 56)(14 49)(15 50)(16 51)(17 110)(18 111)(19 112)(20 105)(21 106)(22 107)(23 108)(24 109)(25 29)(26 30)(27 31)(28 32)(33 75)(34 76)(35 77)(36 78)(37 79)(38 80)(39 73)(40 74)(41 62)(42 63)(43 64)(44 57)(45 58)(46 59)(47 60)(48 61)(65 91)(66 92)(67 93)(68 94)(69 95)(70 96)(71 89)(72 90)(81 85)(82 86)(83 87)(84 88)(97 101)(98 102)(99 103)(100 104)(113 117)(114 118)(115 119)(116 120)
(1 99 88 113 28)(2 29 114 81 100)(3 101 82 115 30)(4 31 116 83 102)(5 103 84 117 32)(6 25 118 85 104)(7 97 86 119 26)(8 27 120 87 98)(9 43 72 39 22)(10 23 40 65 44)(11 45 66 33 24)(12 17 34 67 46)(13 47 68 35 18)(14 19 36 69 48)(15 41 70 37 20)(16 21 38 71 42)(49 112 78 95 61)(50 62 96 79 105)(51 106 80 89 63)(52 64 90 73 107)(53 108 74 91 57)(54 58 92 75 109)(55 110 76 93 59)(56 60 94 77 111)
(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)

G:=sub<Sym(120)| (1,11,50)(2,12,51)(3,13,52)(4,14,53)(5,15,54)(6,16,55)(7,9,56)(8,10,49)(17,106,29)(18,107,30)(19,108,31)(20,109,32)(21,110,25)(22,111,26)(23,112,27)(24,105,28)(33,79,113)(34,80,114)(35,73,115)(36,74,116)(37,75,117)(38,76,118)(39,77,119)(40,78,120)(41,58,103)(42,59,104)(43,60,97)(44,61,98)(45,62,99)(46,63,100)(47,64,101)(48,57,102)(65,95,87)(66,96,88)(67,89,81)(68,90,82)(69,91,83)(70,92,84)(71,93,85)(72,94,86), (1,5)(2,6)(3,7)(4,8)(9,52)(10,53)(11,54)(12,55)(13,56)(14,49)(15,50)(16,51)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,29)(26,30)(27,31)(28,32)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(81,85)(82,86)(83,87)(84,88)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,99,88,113,28)(2,29,114,81,100)(3,101,82,115,30)(4,31,116,83,102)(5,103,84,117,32)(6,25,118,85,104)(7,97,86,119,26)(8,27,120,87,98)(9,43,72,39,22)(10,23,40,65,44)(11,45,66,33,24)(12,17,34,67,46)(13,47,68,35,18)(14,19,36,69,48)(15,41,70,37,20)(16,21,38,71,42)(49,112,78,95,61)(50,62,96,79,105)(51,106,80,89,63)(52,64,90,73,107)(53,108,74,91,57)(54,58,92,75,109)(55,110,76,93,59)(56,60,94,77,111), (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)>;

G:=Group( (1,11,50)(2,12,51)(3,13,52)(4,14,53)(5,15,54)(6,16,55)(7,9,56)(8,10,49)(17,106,29)(18,107,30)(19,108,31)(20,109,32)(21,110,25)(22,111,26)(23,112,27)(24,105,28)(33,79,113)(34,80,114)(35,73,115)(36,74,116)(37,75,117)(38,76,118)(39,77,119)(40,78,120)(41,58,103)(42,59,104)(43,60,97)(44,61,98)(45,62,99)(46,63,100)(47,64,101)(48,57,102)(65,95,87)(66,96,88)(67,89,81)(68,90,82)(69,91,83)(70,92,84)(71,93,85)(72,94,86), (1,5)(2,6)(3,7)(4,8)(9,52)(10,53)(11,54)(12,55)(13,56)(14,49)(15,50)(16,51)(17,110)(18,111)(19,112)(20,105)(21,106)(22,107)(23,108)(24,109)(25,29)(26,30)(27,31)(28,32)(33,75)(34,76)(35,77)(36,78)(37,79)(38,80)(39,73)(40,74)(41,62)(42,63)(43,64)(44,57)(45,58)(46,59)(47,60)(48,61)(65,91)(66,92)(67,93)(68,94)(69,95)(70,96)(71,89)(72,90)(81,85)(82,86)(83,87)(84,88)(97,101)(98,102)(99,103)(100,104)(113,117)(114,118)(115,119)(116,120), (1,99,88,113,28)(2,29,114,81,100)(3,101,82,115,30)(4,31,116,83,102)(5,103,84,117,32)(6,25,118,85,104)(7,97,86,119,26)(8,27,120,87,98)(9,43,72,39,22)(10,23,40,65,44)(11,45,66,33,24)(12,17,34,67,46)(13,47,68,35,18)(14,19,36,69,48)(15,41,70,37,20)(16,21,38,71,42)(49,112,78,95,61)(50,62,96,79,105)(51,106,80,89,63)(52,64,90,73,107)(53,108,74,91,57)(54,58,92,75,109)(55,110,76,93,59)(56,60,94,77,111), (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) );

G=PermutationGroup([(1,11,50),(2,12,51),(3,13,52),(4,14,53),(5,15,54),(6,16,55),(7,9,56),(8,10,49),(17,106,29),(18,107,30),(19,108,31),(20,109,32),(21,110,25),(22,111,26),(23,112,27),(24,105,28),(33,79,113),(34,80,114),(35,73,115),(36,74,116),(37,75,117),(38,76,118),(39,77,119),(40,78,120),(41,58,103),(42,59,104),(43,60,97),(44,61,98),(45,62,99),(46,63,100),(47,64,101),(48,57,102),(65,95,87),(66,96,88),(67,89,81),(68,90,82),(69,91,83),(70,92,84),(71,93,85),(72,94,86)], [(1,5),(2,6),(3,7),(4,8),(9,52),(10,53),(11,54),(12,55),(13,56),(14,49),(15,50),(16,51),(17,110),(18,111),(19,112),(20,105),(21,106),(22,107),(23,108),(24,109),(25,29),(26,30),(27,31),(28,32),(33,75),(34,76),(35,77),(36,78),(37,79),(38,80),(39,73),(40,74),(41,62),(42,63),(43,64),(44,57),(45,58),(46,59),(47,60),(48,61),(65,91),(66,92),(67,93),(68,94),(69,95),(70,96),(71,89),(72,90),(81,85),(82,86),(83,87),(84,88),(97,101),(98,102),(99,103),(100,104),(113,117),(114,118),(115,119),(116,120)], [(1,99,88,113,28),(2,29,114,81,100),(3,101,82,115,30),(4,31,116,83,102),(5,103,84,117,32),(6,25,118,85,104),(7,97,86,119,26),(8,27,120,87,98),(9,43,72,39,22),(10,23,40,65,44),(11,45,66,33,24),(12,17,34,67,46),(13,47,68,35,18),(14,19,36,69,48),(15,41,70,37,20),(16,21,38,71,42),(49,112,78,95,61),(50,62,96,79,105),(51,106,80,89,63),(52,64,90,73,107),(53,108,74,91,57),(54,58,92,75,109),(55,110,76,93,59),(56,60,94,77,111)], [(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)])

S3×C52C8 is a maximal subgroup of
C15⋊M5(2)  S3×C8×D5  C40.34D6  C40.35D6  D12.2Dic5  D12.Dic5  D20.24D6  C60.19C23  D20.27D6  Dic10.27D6
S3×C52C8 is a maximal quotient of
C40.52D6  C60.94D4  C60.15Q8

48 conjugacy classes

class 1 2A2B2C 3 4A4B4C4D5A5B 6 8A8B8C8D8E8F8G8H10A10B10C10D10E10F12A12B15A15B20A20B20C20D20E20F20G20H24A24B24C24D30A30B60A60B60C60D
order1222344445568888888810101010101012121515202020202020202024242424303060606060
size11332113322255551515151522666622442222666610101010444444

48 irreducible representations

dim1111111222222222444
type+++++++-+-+-
imageC1C2C2C2C4C4C8S3D5D6Dic5D10Dic5C4×S3C52C8S3×C8S3×D5S3×Dic5S3×C52C8
kernelS3×C52C8C3×C52C8C153C8S3×C20C5×Dic3S3×C10C5×S3C52C8C4×S3C20Dic3C12D6C10S3C5C4C2C1
# reps1111228121222284224

Matrix representation of S3×C52C8 in GL4(𝔽241) generated by

1000
0100
000240
001240
,
240000
024000
0001
0010
,
5124000
1000
0010
0001
,
2417400
19321700
002400
000240
G:=sub<GL(4,GF(241))| [1,0,0,0,0,1,0,0,0,0,0,1,0,0,240,240],[240,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0],[51,1,0,0,240,0,0,0,0,0,1,0,0,0,0,1],[24,193,0,0,174,217,0,0,0,0,240,0,0,0,0,240] >;

S3×C52C8 in GAP, Magma, Sage, TeX

S_3\times C_5\rtimes_2C_8
% in TeX

G:=Group("S3xC5:2C8");
// GroupNames label

G:=SmallGroup(240,8);
// by ID

G=gap.SmallGroup(240,8);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-3,-5,24,50,490,6917]);
// Polycyclic

G:=Group<a,b,c,d|a^3=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

Export

Subgroup lattice of S3×C52C8 in TeX

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