Copied to
clipboard

G = C2xC5:S4  order 240 = 24·3·5

Direct product of C2 and C5:S4

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

Derived series
C1 — C22 — C5xA4 — C2xC5:S4
Chief series
C1 — C22 — C2xC10 — C5xA4 — C5:S4 — C2xC5:S4
Lower central
C5xA4 — C2xC5:S4
Upper central
C1 — C2

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 12A2B2C2D2E34A4B5A5B610A10B10C10D10E10F15A15B15C15D30A30B30C30D
 size 113330308303022822666688888888
ρ111111111111111111111111111    trivial
ρ21111-1-11-1-111111111111111111    linear of order 2
ρ31-11-11-11-1111-1-1-11-1-111111-1-1-1-1    linear of order 2
ρ41-11-1-1111-111-1-1-11-1-111111-1-1-1-1    linear of order 2
ρ5222200-10022-1222222-1-1-1-1-1-1-1-1    orthogonal lifted from S3
ρ62-22-200-100221-2-22-2-22-1-1-1-11111    orthogonal lifted from D6
ρ72-22-200200-1-√5/2-1+√5/2-21-√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/21+√5/21-√5/21-√5/21+√5/2    orthogonal lifted from D10
ρ8222200200-1+√5/2-1-√5/22-1-√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
ρ9222200200-1-√5/2-1+√5/22-1+√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
ρ102-22-200200-1+√5/2-1-√5/2-21+√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/2-1-√5/2-1+√5/2-1+√5/2-1-√5/21-√5/21+√5/21+√5/21-√5/2    orthogonal lifted from D10
ρ112-22-200-100-1+√5/2-1-√5/211+√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/2ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5+ζ5ζ3ζ53-ζ3ζ52+ζ53-ζ3ζ53+ζ3ζ52+ζ52-ζ3ζ54+ζ3ζ5+ζ5    orthogonal lifted from D30
ρ12222200-100-1-√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-ζ53ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5-ζ54-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53    orthogonal lifted from D15
ρ13222200-100-1+√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-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53-ζ3ζ54+ζ3ζ5-ζ54    orthogonal lifted from D15
ρ142-22-200-100-1-√5/2-1+√5/211-√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-ζ3ζ54+ζ3ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5-ζ54ζ3ζ53-ζ3ζ52+ζ53-ζ3ζ54+ζ3ζ5+ζ5-ζ32ζ54+ζ32ζ5+ζ5-ζ3ζ53+ζ3ζ52+ζ52    orthogonal lifted from D30
ρ152-22-200-100-1+√5/2-1-√5/211+√5/21-√5/2-1-√5/21+√5/21-√5/2-1+√5/2-ζ3ζ53+ζ3ζ52-ζ53-ζ3ζ54+ζ3ζ5-ζ54-ζ32ζ54+ζ32ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5+ζ5-ζ3ζ53+ζ3ζ52+ζ52ζ3ζ53-ζ3ζ52+ζ53-ζ32ζ54+ζ32ζ5+ζ5    orthogonal lifted from D30
ρ16222200-100-1+√5/2-1-√5/2-1-1-√5/2-1+√5/2-1-√5/2-1-√5/2-1+√5/2-1+√5/2ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52-ζ32ζ54+ζ32ζ5-ζ54    orthogonal lifted from D15
ρ17222200-100-1-√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-ζ54ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ54+ζ3ζ5-ζ54ζ3ζ53-ζ3ζ52-ζ52    orthogonal lifted from D15
ρ182-22-200-100-1-√5/2-1+√5/211-√5/21+√5/2-1+√5/21-√5/21+√5/2-1-√5/2-ζ32ζ54+ζ32ζ5-ζ54-ζ3ζ53+ζ3ζ52-ζ53ζ3ζ53-ζ3ζ52-ζ52-ζ3ζ54+ζ3ζ5-ζ54-ζ3ζ53+ζ3ζ52+ζ52-ζ32ζ54+ζ32ζ5+ζ5-ζ3ζ54+ζ3ζ5+ζ5ζ3ζ53-ζ3ζ52+ζ53    orthogonal lifted from D30
ρ1933-1-1-1-101133033-1-1-1-100000000    orthogonal lifted from S4
ρ2033-1-1110-1-133033-1-1-1-100000000    orthogonal lifted from S4
ρ213-3-111-101-1330-3-3-111-100000000    orthogonal lifted from C2xS4
ρ223-3-11-110-11330-3-3-111-100000000    orthogonal lifted from C2xS4
ρ2366-2-200000-3-3√5/2-3+3√5/20-3+3√5/2-3-3√5/21-√5/21-√5/21+√5/21+√5/200000000    orthogonal lifted from C5:S4
ρ2466-2-200000-3+3√5/2-3-3√5/20-3-3√5/2-3+3√5/21+√5/21+√5/21-√5/21-√5/200000000    orthogonal lifted from C5:S4
ρ256-6-2200000-3+3√5/2-3-3√5/203+3√5/23-3√5/21+√5/2-1-√5/2-1+√5/21-√5/200000000    orthogonal faithful
ρ266-6-2200000-3-3√5/2-3+3√5/203-3√5/23+3√5/21-√5/2-1+√5/2-1-√5/21+√5/200000000    orthogonal faithful

Permutation representations of C2xC5:S4
►On 30 points - transitive group 30T61
Generators in S30
(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)

600000
060000
00100
00010
00001
,
1760000
10000
00100
00010
00001
,
10000
01000
006000
00010
000160
,
10000
01000
006000
000600
0059601
,
3628000
3324000
002159
006000
00323259
,
3728000
4724000
00010
00100
00323260

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
 

Export

Character table of C2xC5:S4 in TeX

׿
x
:
Z
F
o
wr
Q
<