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## G = C2×C5⋊S4order 240 = 24·3·5

### Direct product of C2 and C5⋊S4

Aliases: C2×C5⋊S4, C10⋊S4, C23⋊D15, C22⋊D30, A42D10, C52(C2×S4), (C2×A4)⋊D5, (C2×C10)⋊3D6, (C10×A4)⋊1C2, (C5×A4)⋊2C22, (C22×C10)⋊2S3, SmallGroup(240,197)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C2×C5⋊S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — C5⋊S4 — C2×C5⋊S4
 Lower central C5×A4 — C2×C5⋊S4
 Upper central C1 — C2

Generators and relations for C2×C5⋊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, C2×C4, D4, C23, C23, D5, C10, C10, A4, D6, C15, C2×D4, Dic5, D10, C2×C10, C2×C10, S4, C2×A4, D15, C30, C2×Dic5, C5⋊D4, C22×D5, C22×C10, C2×S4, C5×A4, D30, C2×C5⋊D4, C5⋊S4, C10×A4, C2×C5⋊S4
Quotients: C1, C2, C22, S3, D5, D6, D10, S4, D15, C2×S4, D30, C5⋊S4, C2×C5⋊S4

Character table of C2×C5⋊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 C2×S4 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 C2×S4 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

Permutation representations of C2×C5⋊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);

C2×C5⋊S4 is a maximal subgroup of   Dic52S4  Dic5⋊S4  A4⋊D20  C20⋊S4  C242D15  C2×D5×S4
C2×C5⋊S4 is a maximal quotient of   C20.1S4  C20⋊S4  Q8.D30  C20.2S4  C20.6S4  C20.3S4  C242D15

Matrix representation of C2×C5⋊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] >;

C2×C5⋊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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