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G = C2×A5⋊C4  order 480 = 25·3·5

Direct product of C2 and A5⋊C4

direct product, non-abelian, not soluble

Aliases: C2×A5⋊C4, C22.5S5, (C2×A5)⋊C4, A5⋊2(C2×C4), C2.2(C2×S5), (C22×A5).C2, (C2×A5).5C22, SmallGroup(480,952)

Series: Chief►Derived ►Lower central ►Upper central

Chief series
C1 — C2 — C22 — C22×A5 — C2×A5⋊C4
Derived series
A5 — C2×A5⋊C4
Lower central
A5 — C2×A5⋊C4
Upper central
C1 — C22

Subgroups: 1140 in 117 conjugacy classes, 13 normal (7 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, C23, D5, C10, Dic3, C12, A4, D6, C2×C6, C22⋊C4, C22×C4, C24, F5, D10, C2×C10, C4×S3, C2×Dic3, C2×C12, C2×A4, C22×S3, C2×C22⋊C4, C2×F5, C22×D5, A4⋊C4, S3×C2×C4, C22×A4, A5, C22×F5, C2×A4⋊C4, C2×A5, C2×A5, A5⋊C4, C22×A5, C2×A5⋊C4
Quotients: C1, C2, C4, C22, C2×C4, S5, A5⋊C4, C2×S5, C2×A5⋊C4

Character table of C2×A5⋊C4

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H56A6B6C10A10B10C12A12B12C12D
 size 1111151515152010101010303030302420202024242420202020
ρ11111111111111111111111111111    trivial
ρ2111111111-1-1-1-1-1-1-1-11111111-1-1-1-1    linear of order 2
ρ311-1-111-1-11-1-11111-1-111-1-1-1-11-1-111    linear of order 2
ρ411-1-111-1-1111-1-1-1-11111-1-1-1-1111-1-1    linear of order 2
ρ51-11-1-11-111-iii-i-ii-ii1-1-111-1-1i-ii-i    linear of order 4
ρ61-11-1-11-111i-i-iii-ii-i1-1-111-1-1-ii-ii    linear of order 4
ρ71-1-11-111-11-ii-iii-i-ii1-11-1-11-1i-i-ii    linear of order 4
ρ81-1-11-111-11i-ii-i-iii-i1-11-1-11-1-iii-i    linear of order 4
ρ944440000122220000-1111-1-1-1-1-1-1-1    orthogonal lifted from S5
ρ1044-4-400001-2-2220000-11-1-111-111-1-1    orthogonal lifted from C2×S5
ρ11444400001-2-2-2-20000-1111-1-1-11111    orthogonal lifted from S5
ρ1244-4-40000122-2-20000-11-1-111-1-1-111    orthogonal lifted from C2×S5
ρ134-44-400001-2i2i2i-2i0000-1-1-11-111-ii-ii    complex lifted from A5⋊C4
ρ144-4-44000012i-2i2i-2i0000-1-11-11-11i-i-ii    complex lifted from A5⋊C4
ρ154-4-4400001-2i2i-2i2i0000-1-11-11-11-iii-i    complex lifted from A5⋊C4
ρ164-44-4000012i-2i-2i2i0000-1-1-11-111i-ii-i    complex lifted from A5⋊C4
ρ1755-5-511-1-1-111-1-111-1-10-11100011-1-1    orthogonal lifted from C2×S5
ρ1855-5-511-1-1-1-1-111-1-1110-111000-1-111    orthogonal lifted from C2×S5
ρ1955551111-11111-1-1-1-10-1-1-10001111    orthogonal lifted from S5
ρ2055551111-1-1-1-1-111110-1-1-1000-1-1-1-1    orthogonal lifted from S5
ρ215-5-55-111-1-1i-ii-ii-i-ii01-11000-iii-i    complex lifted from A5⋊C4
ρ225-5-55-111-1-1-ii-ii-iii-i01-11000i-i-ii    complex lifted from A5⋊C4
ρ235-55-5-11-11-1i-i-ii-ii-ii011-1000-ii-ii    complex lifted from A5⋊C4
ρ245-55-5-11-11-1-iii-ii-ii-i011-1000i-ii-i    complex lifted from A5⋊C4
ρ256-66-62-22-200000000010001-1-10000    orthogonal lifted from A5⋊C4
ρ266-6-662-2-220000000001000-11-10000    orthogonal lifted from A5⋊C4
ρ276666-2-2-2-200000000010001110000    orthogonal lifted from S5
ρ2866-6-6-2-2220000000001000-1-110000    orthogonal lifted from C2×S5

Permutation representations of C2×A5⋊C4
►On 24 points - transitive group 24T1346
Generators in S24
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)

(1 14 7 20)(2 19 8 13)(3 17 9 23)(4 24 10 18)(5 21 11 15)(6 16 12 22)
 
G:=sub<Sym(24)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24), (1,14,7,20)(2,19,8,13)(3,17,9,23)(4,24,10,18)(5,21,11,15)(6,16,12,22)>;
 
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), (1,14,7,20)(2,19,8,13)(3,17,9,23)(4,24,10,18)(5,21,11,15)(6,16,12,22) );
 
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)], [(1,14,7,20),(2,19,8,13),(3,17,9,23),(4,24,10,18),(5,21,11,15),(6,16,12,22)]])
 
G:=TransitiveGroup(24,1346);
 

Matrix representation of C2×A5⋊C4 ►in GL6(𝔽61)

1100000
0500000
001000
0010060
0016000
0010600
,
5000000
0500000
0006000
0060000
0000600
0000060

G:=sub<GL(6,GF(61))| [11,0,0,0,0,0,0,50,0,0,0,0,0,0,1,1,1,1,0,0,0,0,60,0,0,0,0,0,0,60,0,0,0,60,0,0],[50,0,0,0,0,0,0,50,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,60] >;
 

C2×A5⋊C4 in GAP, Magma, Sage, TeX 

C_2\times A_5\rtimes C_4
 
% in TeX
 
G:=Group("C2xA5:C4");
 
// GroupNames label
 
G:=SmallGroup(480,952);
 
// by ID
 
G=gap.SmallGroup(480,952);
 
# by ID
 

Export

Character table of C2×A5⋊C4 in TeX

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