direct product, non-abelian, soluble, monomial
Aliases: C2xC5:S4, C10:S4, C23:D15, C22:D30, A4:2D10, C5:2(C2xS4), (C2xA4):D5, (C2xC10):3D6, (C10xA4):1C2, (C5xA4):2C22, (C22xC10):2S3, SmallGroup(240,197)
Series: Derived ►Chief ►Lower central ►Upper central
C5xA4 — C2xC5:S4 |
Generators and relations for C2xC5:S4
G = < a,b,c,d,e,f | a2=b5=c2=d2=e3=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf=b-1, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >
Subgroups: 484 in 66 conjugacy classes, 15 normal (13 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2xC4, D4, C23, C23, D5, C10, C10, A4, D6, C15, C2xD4, Dic5, D10, C2xC10, C2xC10, S4, C2xA4, D15, C30, C2xDic5, C5:D4, C22xD5, C22xC10, C2xS4, C5xA4, D30, C2xC5:D4, C5:S4, C10xA4, C2xC5:S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2xS4, D30, C5:S4, C2xC5:S4
Character table of C2xC5:S4
class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 5A | 5B | 6 | 10A | 10B | 10C | 10D | 10E | 10F | 15A | 15B | 15C | 15D | 30A | 30B | 30C | 30D | |
size | 1 | 1 | 3 | 3 | 30 | 30 | 8 | 30 | 30 | 2 | 2 | 8 | 2 | 2 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | |
ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
ρ2 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
ρ3 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ4 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
ρ5 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
ρ6 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | 2 | 2 | 1 | -2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
ρ7 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | -2 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -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 D10 |
ρ8 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | 2 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -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 |
ρ9 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 0 | 0 | -1-√5/2 | -1+√5/2 | 2 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -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 |
ρ10 | 2 | -2 | 2 | -2 | 0 | 0 | 2 | 0 | 0 | -1+√5/2 | -1-√5/2 | -2 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -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 D10 |
ρ11 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5+ζ5 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ53+ζ3ζ52+ζ52 | -ζ3ζ54+ζ3ζ5+ζ5 | orthogonal lifted from D30 |
ρ12 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | orthogonal lifted from D15 |
ρ13 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | orthogonal lifted from D15 |
ρ14 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ3ζ54+ζ3ζ5+ζ5 | -ζ32ζ54+ζ32ζ5+ζ5 | -ζ3ζ53+ζ3ζ52+ζ52 | orthogonal lifted from D30 |
ρ15 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | 1 | 1+√5/2 | 1-√5/2 | -1-√5/2 | 1+√5/2 | 1-√5/2 | -1+√5/2 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ32ζ54+ζ32ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5+ζ5 | -ζ3ζ53+ζ3ζ52+ζ52 | ζ3ζ53-ζ3ζ52+ζ53 | -ζ32ζ54+ζ32ζ5+ζ5 | orthogonal lifted from D30 |
ρ16 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1+√5/2 | -1-√5/2 | -1 | -1-√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ32ζ54+ζ32ζ5-ζ54 | orthogonal lifted from D15 |
ρ17 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | -1 | -1+√5/2 | -1-√5/2 | -1+√5/2 | -1+√5/2 | -1-√5/2 | -1-√5/2 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ54+ζ3ζ5-ζ54 | ζ3ζ53-ζ3ζ52-ζ52 | orthogonal lifted from D15 |
ρ18 | 2 | -2 | 2 | -2 | 0 | 0 | -1 | 0 | 0 | -1-√5/2 | -1+√5/2 | 1 | 1-√5/2 | 1+√5/2 | -1+√5/2 | 1-√5/2 | 1+√5/2 | -1-√5/2 | -ζ32ζ54+ζ32ζ5-ζ54 | -ζ3ζ53+ζ3ζ52-ζ53 | ζ3ζ53-ζ3ζ52-ζ52 | -ζ3ζ54+ζ3ζ5-ζ54 | -ζ3ζ53+ζ3ζ52+ζ52 | -ζ32ζ54+ζ32ζ5+ζ5 | -ζ3ζ54+ζ3ζ5+ζ5 | ζ3ζ53-ζ3ζ52+ζ53 | orthogonal lifted from D30 |
ρ19 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 1 | 1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ20 | 3 | 3 | -1 | -1 | 1 | 1 | 0 | -1 | -1 | 3 | 3 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S4 |
ρ21 | 3 | -3 | -1 | 1 | 1 | -1 | 0 | 1 | -1 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ22 | 3 | -3 | -1 | 1 | -1 | 1 | 0 | -1 | 1 | 3 | 3 | 0 | -3 | -3 | -1 | 1 | 1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2xS4 |
ρ23 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | -3+3√5/2 | -3-3√5/2 | 1-√5/2 | 1-√5/2 | 1+√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5:S4 |
ρ24 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | -3-3√5/2 | -3+3√5/2 | 1+√5/2 | 1+√5/2 | 1-√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C5:S4 |
ρ25 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3+3√5/2 | -3-3√5/2 | 0 | 3+3√5/2 | 3-3√5/2 | 1+√5/2 | -1-√5/2 | -1+√5/2 | 1-√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
ρ26 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -3-3√5/2 | -3+3√5/2 | 0 | 3-3√5/2 | 3+3√5/2 | 1-√5/2 | -1+√5/2 | -1-√5/2 | 1+√5/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
(1 13)(2 14)(3 15)(4 11)(5 12)(6 29)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)(20 25)
(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 13)(2 14)(3 15)(4 11)(5 12)(6 29)(7 30)(8 26)(9 27)(10 28)
(6 29)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 24 14)(7 25 15)(8 21 11)(9 22 12)(10 23 13)
(2 5)(3 4)(6 22)(7 21)(8 25)(9 24)(10 23)(11 15)(12 14)(16 30)(17 29)(18 28)(19 27)(20 26)
G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28), (6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,24,14)(7,25,15)(8,21,11)(9,22,12)(10,23,13), (2,5)(3,4)(6,22)(7,21)(8,25)(9,24)(10,23)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26)>;
G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (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), (1,13)(2,14)(3,15)(4,11)(5,12)(6,29)(7,30)(8,26)(9,27)(10,28), (6,29)(7,30)(8,26)(9,27)(10,28)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,24,14)(7,25,15)(8,21,11)(9,22,12)(10,23,13), (2,5)(3,4)(6,22)(7,21)(8,25)(9,24)(10,23)(11,15)(12,14)(16,30)(17,29)(18,28)(19,27)(20,26) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,11),(5,12),(6,29),(7,30),(8,26),(9,27),(10,28),(16,21),(17,22),(18,23),(19,24),(20,25)], [(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)], [(1,13),(2,14),(3,15),(4,11),(5,12),(6,29),(7,30),(8,26),(9,27),(10,28)], [(6,29),(7,30),(8,26),(9,27),(10,28),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,24,14),(7,25,15),(8,21,11),(9,22,12),(10,23,13)], [(2,5),(3,4),(6,22),(7,21),(8,25),(9,24),(10,23),(11,15),(12,14),(16,30),(17,29),(18,28),(19,27),(20,26)]])
G:=TransitiveGroup(30,61);
C2xC5:S4 is a maximal subgroup of
Dic5:2S4 Dic5:S4 A4:D20 C20:S4 C24:2D15 C2xD5xS4
C2xC5:S4 is a maximal quotient of C20.1S4 C20:S4 Q8.D30 C20.2S4 C20.6S4 C20.3S4 C24:2D15
Matrix representation of C2xC5:S4 ►in GL5(F61)
60 | 0 | 0 | 0 | 0 |
0 | 60 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
17 | 60 | 0 | 0 | 0 |
1 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 0 | 1 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 60 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 0 | 1 | 60 |
1 | 0 | 0 | 0 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 0 | 60 | 0 | 0 |
0 | 0 | 0 | 60 | 0 |
0 | 0 | 59 | 60 | 1 |
36 | 28 | 0 | 0 | 0 |
33 | 24 | 0 | 0 | 0 |
0 | 0 | 2 | 1 | 59 |
0 | 0 | 60 | 0 | 0 |
0 | 0 | 32 | 32 | 59 |
37 | 28 | 0 | 0 | 0 |
47 | 24 | 0 | 0 | 0 |
0 | 0 | 0 | 1 | 0 |
0 | 0 | 1 | 0 | 0 |
0 | 0 | 32 | 32 | 60 |
G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[17,1,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,1,1,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,59,0,0,0,60,60,0,0,0,0,1],[36,33,0,0,0,28,24,0,0,0,0,0,2,60,32,0,0,1,0,32,0,0,59,0,59],[37,47,0,0,0,28,24,0,0,0,0,0,0,1,32,0,0,1,0,32,0,0,0,0,60] >;
C2xC5:S4 in GAP, Magma, Sage, TeX
C_2\times C_5\rtimes S_4
% in TeX
G:=Group("C2xC5:S4");
// GroupNames label
G:=SmallGroup(240,197);
// by ID
G=gap.SmallGroup(240,197);
# by ID
G:=PCGroup([6,-2,-2,-3,-5,-2,2,146,1155,3604,916,2165,1637]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^2=b^5=c^2=d^2=e^3=f^2=1,a*b=b*a,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,f*b*f=b^-1,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
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