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G = C5×C8⋊S3order 240 = 24·3·5

Direct product of C5 and C8⋊S3

direct product, metacyclic, supersoluble, monomial, 2-hyperelementary

Aliases: C5×C8⋊S3, C407S3, D6.C20, C245C10, C12013C2, C20.57D6, Dic3.C20, C1512M4(2), C60.74C22, C3⋊C84C10, C83(C5×S3), C2.3(S3×C20), C6.2(C2×C20), C31(C5×M4(2)), (C4×S3).2C10, (S3×C20).5C2, (S3×C10).5C4, C10.23(C4×S3), C4.13(S3×C10), C30.46(C2×C4), C12.13(C2×C10), (C5×Dic3).5C4, (C5×C3⋊C8)⋊11C2, SmallGroup(240,50)

Series: Derived Chief Lower central Upper central

C1C6 — C5×C8⋊S3
C1C3C6C12C60S3×C20 — C5×C8⋊S3
C3C6 — C5×C8⋊S3
C1C20C40

Generators and relations for C5×C8⋊S3
 G = < a,b,c,d | a5=b8=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, dbd=b5, dcd=c-1 >

6C2
3C22
3C4
2S3
6C10
3C2×C4
3C8
3C20
3C2×C10
2C5×S3
3M4(2)
3C40
3C2×C20
3C5×M4(2)

Smallest permutation representation of C5×C8⋊S3
On 120 points
Generators in S120
(1 15 26 51 24)(2 16 27 52 17)(3 9 28 53 18)(4 10 29 54 19)(5 11 30 55 20)(6 12 31 56 21)(7 13 32 49 22)(8 14 25 50 23)(33 60 90 107 83)(34 61 91 108 84)(35 62 92 109 85)(36 63 93 110 86)(37 64 94 111 87)(38 57 95 112 88)(39 58 96 105 81)(40 59 89 106 82)(41 117 76 101 67)(42 118 77 102 68)(43 119 78 103 69)(44 120 79 104 70)(45 113 80 97 71)(46 114 73 98 72)(47 115 74 99 65)(48 116 75 100 66)
(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)
(1 103 82)(2 104 83)(3 97 84)(4 98 85)(5 99 86)(6 100 87)(7 101 88)(8 102 81)(9 71 34)(10 72 35)(11 65 36)(12 66 37)(13 67 38)(14 68 39)(15 69 40)(16 70 33)(17 79 107)(18 80 108)(19 73 109)(20 74 110)(21 75 111)(22 76 112)(23 77 105)(24 78 106)(25 42 58)(26 43 59)(27 44 60)(28 45 61)(29 46 62)(30 47 63)(31 48 64)(32 41 57)(49 117 95)(50 118 96)(51 119 89)(52 120 90)(53 113 91)(54 114 92)(55 115 93)(56 116 94)
(2 6)(4 8)(10 14)(12 16)(17 21)(19 23)(25 29)(27 31)(33 66)(34 71)(35 68)(36 65)(37 70)(38 67)(39 72)(40 69)(41 57)(42 62)(43 59)(44 64)(45 61)(46 58)(47 63)(48 60)(50 54)(52 56)(73 105)(74 110)(75 107)(76 112)(77 109)(78 106)(79 111)(80 108)(81 98)(82 103)(83 100)(84 97)(85 102)(86 99)(87 104)(88 101)(89 119)(90 116)(91 113)(92 118)(93 115)(94 120)(95 117)(96 114)

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

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

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

C5×C8⋊S3 is a maximal subgroup of
C40⋊D6  C401D6  D40⋊S3  Dic20⋊S3  C40.34D6  C40.35D6  C40.2D6  C5×S3×M4(2)

90 conjugacy classes

class 1 2A2B 3 4A4B4C5A5B5C5D 6 8A8B8C8D10A10B10C10D10E10F10G10H12A12B15A15B15C15D20A···20H20I20J20K20L24A24B24C24D30A30B30C30D40A···40H40I···40P60A···60H120A···120P
order1223444555568888101010101010101012121515151520···2020202020242424243030303040···4040···4060···60120···120
size1162116111122266111166662222221···16666222222222···26···62···22···2

90 irreducible representations

dim1111111111112222222222
type++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20S3D6M4(2)C4×S3C5×S3C8⋊S3S3×C10C5×M4(2)S3×C20C5×C8⋊S3
kernelC5×C8⋊S3C5×C3⋊C8C120S3×C20C5×Dic3S3×C10C8⋊S3C3⋊C8C24C4×S3Dic3D6C40C20C15C10C8C5C4C3C2C1
# reps11112244448811224448816

Matrix representation of C5×C8⋊S3 in GL4(𝔽241) generated by

91000
09100
00980
00098
,
012000
113000
0010
0001
,
1000
0100
0001
00240240
,
1000
024000
0001
0010
G:=sub<GL(4,GF(241))| [91,0,0,0,0,91,0,0,0,0,98,0,0,0,0,98],[0,113,0,0,120,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,240,0,0,1,240],[1,0,0,0,0,240,0,0,0,0,0,1,0,0,1,0] >;

C5×C8⋊S3 in GAP, Magma, Sage, TeX

C_5\times C_8\rtimes S_3
% in TeX

G:=Group("C5xC8:S3");
// GroupNames label

G:=SmallGroup(240,50);
// by ID

G=gap.SmallGroup(240,50);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-3,505,127,69,5765]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^8=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

Export

Subgroup lattice of C5×C8⋊S3 in TeX

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