Copied to
clipboard

G = D5xS4order 240 = 24·3·5

Direct product of D5 and S4

direct product, non-abelian, soluble, monomial

Aliases: D5xS4, A4:1D10, C5:S4:C2, (C5xS4):C2, C5:1(C2xS4), (D5xA4):C2, (C2xC10):D6, C22:(S3xD5), (C5xA4):C22, (C22xD5):S3, SmallGroup(240,194)

Series: Derived Chief Lower central Upper central

C1C22C5xA4 — D5xS4
C1C22C2xC10C5xA4D5xA4 — D5xS4
C5xA4 — D5xS4
C1

Generators and relations for D5xS4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e3=f2=1, bab=a-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 468 in 66 conjugacy classes, 13 normal (all characteristic)
C1, C2, C3, C4, C22, C22, C5, S3, C6, C2xC4, D4, C23, D5, D5, C10, A4, D6, C15, C2xD4, Dic5, C20, D10, C2xC10, C2xC10, S4, S4, C2xA4, C5xS3, C3xD5, D15, C4xD5, D20, C5:D4, C5xD4, C22xD5, C22xD5, C2xS4, S3xD5, C5xA4, D4xD5, C5xS4, C5:S4, D5xA4, D5xS4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, C2xS4, S3xD5, D5xS4

Character table of D5xS4

 class 12A2B2C2D2E34A4B5A5B610A10B10C10D15A15B20A20B
 size 135615308630224066121216161212
ρ111111111111111111111    trivial
ρ211-11-1-111-111-111111111    linear of order 2
ρ311-1-1-111-1111-111-1-111-1-1    linear of order 2
ρ4111-11-11-1-111111-1-111-1-1    linear of order 2
ρ522-20-20-1002212200-1-100    orthogonal lifted from D6
ρ6222020-10022-12200-1-100    orthogonal lifted from S3
ρ7220-2002-20-1+5/2-1-5/20-1+5/2-1-5/21-5/21+5/2-1+5/2-1-5/21-5/21+5/2    orthogonal lifted from D10
ρ8220200220-1-5/2-1+5/20-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2    orthogonal lifted from D5
ρ9220200220-1+5/2-1-5/20-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2-1+5/2-1-5/2    orthogonal lifted from D5
ρ10220-2002-20-1-5/2-1+5/20-1-5/2-1+5/21+5/21-5/2-1-5/2-1+5/21+5/21-5/2    orthogonal lifted from D10
ρ113-1-311-10-11330-1-11100-1-1    orthogonal lifted from C2xS4
ρ123-131-110-1-1330-1-11100-1-1    orthogonal lifted from S4
ρ133-13-1-1-1011330-1-1-1-10011    orthogonal lifted from S4
ρ143-1-3-11101-1330-1-1-1-10011    orthogonal lifted from C2xS4
ρ15440000-200-1-5-1+50-1-5-1+5001+5/21-5/200    orthogonal lifted from S3xD5
ρ16440000-200-1+5-1-50-1+5-1-5001-5/21+5/200    orthogonal lifted from S3xD5
ρ176-202000-20-3-35/2-3+35/201+5/21-5/2-1-5/2-1+5/2001+5/21-5/2    orthogonal faithful
ρ186-20-200020-3+35/2-3-35/201-5/21+5/21-5/21+5/200-1+5/2-1-5/2    orthogonal faithful
ρ196-202000-20-3+35/2-3-35/201-5/21+5/2-1+5/2-1-5/2001-5/21+5/2    orthogonal faithful
ρ206-20-200020-3-35/2-3+35/201+5/21-5/21+5/21-5/200-1-5/2-1+5/2    orthogonal faithful

Permutation representations of D5xS4
On 20 points - transitive group 20T69
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 7)(8 10)(11 13)(14 15)(16 20)(17 19)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 16)(2 17)(3 18)(4 19)(5 20)(6 14)(7 15)(8 11)(9 12)(10 13)
(6 20 14)(7 16 15)(8 17 11)(9 18 12)(10 19 13)
(6 14)(7 15)(8 11)(9 12)(10 13)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,7)(8,10)(11,13)(14,15)(16,20)(17,19), (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,16)(2,17)(3,18)(4,19)(5,20)(6,14)(7,15)(8,11)(9,12)(10,13), (6,20,14)(7,16,15)(8,17,11)(9,18,12)(10,19,13), (6,14)(7,15)(8,11)(9,12)(10,13)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,5)(2,4)(6,7)(8,10)(11,13)(14,15)(16,20)(17,19), (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,16)(2,17)(3,18)(4,19)(5,20)(6,14)(7,15)(8,11)(9,12)(10,13), (6,20,14)(7,16,15)(8,17,11)(9,18,12)(10,19,13), (6,14)(7,15)(8,11)(9,12)(10,13) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,5),(2,4),(6,7),(8,10),(11,13),(14,15),(16,20),(17,19)], [(1,7),(2,8),(3,9),(4,10),(5,6),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,16),(2,17),(3,18),(4,19),(5,20),(6,14),(7,15),(8,11),(9,12),(10,13)], [(6,20,14),(7,16,15),(8,17,11),(9,18,12),(10,19,13)], [(6,14),(7,15),(8,11),(9,12),(10,13)]])

G:=TransitiveGroup(20,69);

On 30 points - transitive group 30T54
Generators in S30
(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)
(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)(21 30)(22 26)(23 27)(24 28)(25 29)

G:=sub<Sym(30)| (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29)>;

G:=Group( (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29) );

G=PermutationGroup([[(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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,17),(18,20),(21,24),(22,23),(26,27),(28,30)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19),(21,30),(22,26),(23,27),(24,28),(25,29)]])

G:=TransitiveGroup(30,54);

On 30 points - transitive group 30T59
Generators in S30
(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)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 28)(7 27)(8 26)(9 30)(10 29)(16 23)(17 22)(18 21)(19 25)(20 24)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

G:=sub<Sym(30)| (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30)>;

G:=Group( (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30) );

G=PermutationGroup([[(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,28),(7,27),(8,26),(9,30),(10,29),(16,23),(17,22),(18,21),(19,25),(20,24)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,26),(17,27),(18,28),(19,29),(20,30)]])

G:=TransitiveGroup(30,59);

On 30 points - transitive group 30T62
Generators in S30
(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)
(1 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

G:=sub<Sym(30)| (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30)>;

G:=Group( (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), (1,5)(2,4)(6,9)(7,8)(11,14)(12,13)(16,17)(18,20)(21,24)(22,23)(26,27)(28,30), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (6,21)(7,22)(8,23)(9,24)(10,25)(16,26)(17,27)(18,28)(19,29)(20,30) );

G=PermutationGroup([[(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)], [(1,5),(2,4),(6,9),(7,8),(11,14),(12,13),(16,17),(18,20),(21,24),(22,23),(26,27),(28,30)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(6,21),(7,22),(8,23),(9,24),(10,25),(16,26),(17,27),(18,28),(19,29),(20,30)]])

G:=TransitiveGroup(30,62);

On 30 points - transitive group 30T63
Generators in S30
(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)
(1 12)(2 11)(3 15)(4 14)(5 13)(6 28)(7 27)(8 26)(9 30)(10 29)(16 23)(17 22)(18 21)(19 25)(20 24)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)(21 30)(22 26)(23 27)(24 28)(25 29)

G:=sub<Sym(30)| (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29)>;

G:=Group( (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), (1,12)(2,11)(3,15)(4,14)(5,13)(6,28)(7,27)(8,26)(9,30)(10,29)(16,23)(17,22)(18,21)(19,25)(20,24), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19)(21,30)(22,26)(23,27)(24,28)(25,29) );

G=PermutationGroup([[(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)], [(1,12),(2,11),(3,15),(4,14),(5,13),(6,28),(7,27),(8,26),(9,30),(10,29),(16,23),(17,22),(18,21),(19,25),(20,24)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,20),(7,16),(8,17),(9,18),(10,19),(21,30),(22,26),(23,27),(24,28),(25,29)]])

G:=TransitiveGroup(30,63);

D5xS4 is a maximal quotient of
CSU2(F3):D5  Dic5.6S4  Dic5.7S4  GL2(F3):D5  D10.1S4  D10.2S4  A4:Dic10  Dic5:2S4  Dic5:S4  D10:S4  A4:D20

Matrix representation of D5xS4 in GL5(F61)

01000
6017000
00100
00010
00001
,
060000
600000
006000
000600
000060
,
10000
01000
000601
000600
001600
,
10000
01000
000160
001060
000060
,
10000
01000
00001
00100
00010
,
600000
060000
00010
00100
00001

G:=sub<GL(5,GF(61))| [0,60,0,0,0,1,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,60,0,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,60,60,60,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,60,60,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[60,0,0,0,0,0,60,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,1] >;

D5xS4 in GAP, Magma, Sage, TeX

D_5\times S_4
% in TeX

G:=Group("D5xS4");
// GroupNames label

G:=SmallGroup(240,194);
// by ID

G=gap.SmallGroup(240,194);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,2,80,1155,1810,916,1091,1637]);
// Polycyclic

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

Export

Character table of D5xS4 in TeX

׿
x
:
Z
F
o
wr
Q
<