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G = C4.3S5order 480 = 25·3·5

3rd non-split extension by C4 of S5 acting via S5/A5=C2

non-abelian, not soluble

Aliases: C4.3S5, SL2(𝔽5)⋊1C22, C4.A53C2, C2.7(C2×S5), C2.S51C2, SmallGroup(480,948)

Series: ChiefDerived Lower central Upper central

C1C2C4C4.A5 — C4.3S5
SL2(𝔽5) — C4.3S5
SL2(𝔽5) — C4.3S5
C1C2C4

Subgroups: 934 in 76 conjugacy classes, 8 normal (6 characteristic)
C1, C2, C2 [×3], C3, C4, C4, C22 [×5], C5, S3 [×4], C6 [×3], C8 [×2], C2×C4, D4 [×4], Q8, C23, D5, C10, Dic3, C12, D6 [×7], C2×C6 [×2], M4(2), D8 [×2], SD16 [×2], C2×D4, C4○D4, Dic5, C20, D10, SL2(𝔽3), C4×S3, D12, C3⋊D4 [×2], C3×D4, C22×S3 [×2], C8⋊C22, C5⋊C8 [×2], C4×D5, GL2(𝔽3) [×2], C4.A4, S3×D4, C4.F5, C4.3S4, SL2(𝔽5), C2.S5 [×2], C4.A5, C4.3S5
Quotients: C1, C2 [×3], C22, S5, C2×S5, C4.3S5

Character table of C4.3S5

 class 12A2B2C2D34A4B56A6B6C8A8B101220A20B
 size 112020302023024204040606024402424
ρ1111111111111111111    trivial
ρ211-1-1111111-1-1-1-11111    linear of order 2
ρ311-11-11-1111-11-111-1-1-1    linear of order 2
ρ4111-1-11-11111-11-11-1-1-1    linear of order 2
ρ544220140-11-1-100-11-1-1    orthogonal lifted from S5
ρ6442-201-40-11-1100-1-111    orthogonal lifted from C2×S5
ρ744-2-20140-111100-11-1-1    orthogonal lifted from S5
ρ844-2201-40-111-100-1-111    orthogonal lifted from C2×S5
ρ94-4000-200-12000010-5--5    complex faithful
ρ104-4000-200-12000010--5-5    complex faithful
ρ1155-11-1-1-510-1-111-10100    orthogonal lifted from C2×S5
ρ1255111-1510-111-1-10-100    orthogonal lifted from S5
ρ1355-1-11-1510-1-1-1110-100    orthogonal lifted from S5
ρ14551-1-1-1-510-11-1-110100    orthogonal lifted from C2×S5
ρ15660020-6-210000010-1-1    orthogonal lifted from C2×S5
ρ166600-206-21000001011    orthogonal lifted from S5
ρ178-8000200-2-200002000    orthogonal faithful
ρ1812-12000000200000-2000    orthogonal faithful

Smallest permutation representation of C4.3S5
On 40 points
Generators in S40
(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)
(1 3 5 7)(2 23 6 19)(4 28 8 32)(9 22 13 18)(10 29 14 25)(11 39 15 35)(12 34 16 38)(17 30 21 26)(20 37 24 33)(27 40 31 36)
(1 5)(3 7)(9 38)(10 35)(11 40)(12 37)(13 34)(14 39)(15 36)(16 33)(17 28)(18 25)(19 30)(20 27)(21 32)(22 29)(23 26)(24 31)

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

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

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

Matrix representation of C4.3S5 in GL4(𝔽3) generated by

2101
0210
0100
2202
,
2211
1020
2011
0220
,
1000
1200
2012
1002
G:=sub<GL(4,GF(3))| [2,0,0,2,1,2,1,2,0,1,0,0,1,0,0,2],[2,1,2,0,2,0,0,2,1,2,1,2,1,0,1,0],[1,1,2,1,0,2,0,0,0,0,1,0,0,0,2,2] >;

C4.3S5 in GAP, Magma, Sage, TeX

C_4._3S_5
% in TeX

G:=Group("C4.3S5");
// GroupNames label

G:=SmallGroup(480,948);
// by ID

G=gap.SmallGroup(480,948);
# by ID

Export

Character table of C4.3S5 in TeX

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