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G = C4.S5order 480 = 25·3·5

2nd non-split extension by C4 of S5 acting via S5/A5=C2

non-abelian, not soluble

Aliases: C4.2S5, CSU2(𝔽5)⋊1C2, SL2(𝔽5).2C22, C2.6(C2×S5), C4.A5.1C2, SmallGroup(480,947)

Series: ChiefDerived Lower central Upper central

C1C2C4C4.A5 — C4.S5
SL2(𝔽5) — C4.S5
SL2(𝔽5) — C4.S5
C1C2C4

Subgroups: 614 in 64 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2, C3, C4, C4 [×3], C22, C5, S3 [×2], C6, C8 [×2], C2×C4 [×2], D4, Q8 [×4], D5, C10, Dic3 [×3], C12 [×3], D6, M4(2), SD16 [×2], Q16 [×2], C2×Q8, C4○D4, Dic5, C20, D10, SL2(𝔽3), Dic6 [×3], C4×S3 [×3], C3×Q8, C8.C22, C5⋊C8 [×2], C4×D5, CSU2(𝔽3) [×2], C4.A4, S3×Q8, C4.F5, C4.S4, SL2(𝔽5), CSU2(𝔽5) [×2], C4.A5, C4.S5
Quotients: C1, C2 [×3], C22, S5, C2×S5, C4.S5

Character table of C4.S5

 class 12A2B34A4B4C4D568A8B1012A12B12C20A20B
 size 113020220203024206060244040402424
ρ1111111111111111111    trivial
ρ211-11-1-11111-111-11-1-1-1    linear of order 2
ρ311111-1-1111-1-11-1-1111    linear of order 2
ρ411-11-11-11111-111-1-1-1-1    linear of order 2
ρ54401-42-20-1100-1-11-111    orthogonal lifted from C2×S5
ρ64401-4-220-1100-11-1-111    orthogonal lifted from C2×S5
ρ744014220-1100-1-1-11-1-1    orthogonal lifted from S5
ρ844014-2-20-1100-1111-1-1    orthogonal lifted from S5
ρ94-40-20000-12001000--5-5    complex faithful
ρ104-40-20000-12001000-5--5    complex faithful
ρ11551-151110-1-1-1011-100    orthogonal lifted from S5
ρ1255-1-1-5-1110-11-10-11100    orthogonal lifted from C2×S5
ρ1355-1-1-51-110-1-1101-1100    orthogonal lifted from C2×S5
ρ14551-15-1-110-1110-1-1-100    orthogonal lifted from S5
ρ156620-600-210001000-1-1    orthogonal lifted from C2×S5
ρ1666-20600-21000100011    orthogonal lifted from S5
ρ178-8020000-2-200200000    symplectic faithful, Schur index 2
ρ1812-120000002000-200000    symplectic faithful, Schur index 2

Smallest permutation representation of C4.S5
On 48 points
Generators in S48
(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)
(1 39 5 35)(2 18 6 22)(3 8 7 4)(9 17 13 21)(10 15 14 11)(12 36 16 40)(19 30 23 26)(20 45 24 41)(25 38 29 34)(27 32 31 28)(33 48 37 44)(42 47 46 43)
(1 44 5 48)(2 41 6 45)(3 46 7 42)(4 43 8 47)(9 30 13 26)(10 27 14 31)(11 32 15 28)(12 29 16 25)(17 19 21 23)(18 24 22 20)(33 39 37 35)(34 36 38 40)

G:=sub<Sym(48)| (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), (1,39,5,35)(2,18,6,22)(3,8,7,4)(9,17,13,21)(10,15,14,11)(12,36,16,40)(19,30,23,26)(20,45,24,41)(25,38,29,34)(27,32,31,28)(33,48,37,44)(42,47,46,43), (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,30,13,26)(10,27,14,31)(11,32,15,28)(12,29,16,25)(17,19,21,23)(18,24,22,20)(33,39,37,35)(34,36,38,40)>;

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)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,39,5,35)(2,18,6,22)(3,8,7,4)(9,17,13,21)(10,15,14,11)(12,36,16,40)(19,30,23,26)(20,45,24,41)(25,38,29,34)(27,32,31,28)(33,48,37,44)(42,47,46,43), (1,44,5,48)(2,41,6,45)(3,46,7,42)(4,43,8,47)(9,30,13,26)(10,27,14,31)(11,32,15,28)(12,29,16,25)(17,19,21,23)(18,24,22,20)(33,39,37,35)(34,36,38,40) );

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),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,39,5,35),(2,18,6,22),(3,8,7,4),(9,17,13,21),(10,15,14,11),(12,36,16,40),(19,30,23,26),(20,45,24,41),(25,38,29,34),(27,32,31,28),(33,48,37,44),(42,47,46,43)], [(1,44,5,48),(2,41,6,45),(3,46,7,42),(4,43,8,47),(9,30,13,26),(10,27,14,31),(11,32,15,28),(12,29,16,25),(17,19,21,23),(18,24,22,20),(33,39,37,35),(34,36,38,40)])

Matrix representation of C4.S5 in GL4(𝔽3) generated by

0012
2000
0222
0211
,
2202
0212
2001
2222
,
0011
0012
1100
1200
G:=sub<GL(4,GF(3))| [0,2,0,0,0,0,2,2,1,0,2,1,2,0,2,1],[2,0,2,2,2,2,0,2,0,1,0,2,2,2,1,2],[0,0,1,1,0,0,1,2,1,1,0,0,1,2,0,0] >;

C4.S5 in GAP, Magma, Sage, TeX

C_4.S_5
% in TeX

G:=Group("C4.S5");
// GroupNames label

G:=SmallGroup(480,947);
// by ID

G=gap.SmallGroup(480,947);
# by ID

Export

Character table of C4.S5 in TeX

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