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G = C2×SL2(𝔽5)  order 240 = 24·3·5

Direct product of C2 and SL2(𝔽5)

direct product, non-abelian, not soluble

Aliases: C2×SL2(𝔽5), C22.A5, C2.3(C2×A5), SmallGroup(240,94)

Series: ChiefDerived Lower central Upper central

C1C2C22 — C2×SL2(𝔽5)
SL2(𝔽5) — C2×SL2(𝔽5)
SL2(𝔽5) — C2×SL2(𝔽5)
C1C22

10C3
6C5
15C4
15C4
10C6
10C6
10C6
6C10
6C10
6C10
5Q8
15Q8
15C2×C4
10Dic3
10C2×C6
10Dic3
6Dic5
6C2×C10
6Dic5
5C2×Q8
5SL2(𝔽3)
10C2×Dic3
6C2×Dic5
5C2×SL2(𝔽3)

Character table of C2×SL2(𝔽5)

 class 12A2B2C34A4B5A5B6A6B6C10A10B10C10D10E10F
 size 11112030301212202020121212121212
ρ1111111111111111111    trivial
ρ21-11-111-1111-1-1-1-1-1-111    linear of order 2
ρ322-2-2-100-1+5/2-1-5/211-1-1-5/2-1+5/21-5/21+5/21-5/21+5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ42-2-22-100-1+5/2-1-5/21-111+5/21-5/2-1+5/2-1-5/21-5/21+5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ52-2-22-100-1-5/2-1+5/21-111-5/21+5/2-1-5/2-1+5/21+5/21-5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ622-2-2-100-1-5/2-1+5/211-1-1+5/2-1-5/21+5/21-5/21+5/21-5/2    symplectic lifted from SL2(𝔽5), Schur index 2
ρ73-33-30-111+5/21-5/2000-1+5/2-1-5/2-1-5/2-1+5/21+5/21-5/2    orthogonal lifted from C2×A5
ρ83-33-30-111-5/21+5/2000-1-5/2-1+5/2-1+5/2-1-5/21-5/21+5/2    orthogonal lifted from C2×A5
ρ933330-1-11-5/21+5/20001+5/21-5/21-5/21+5/21-5/21+5/2    orthogonal lifted from A5
ρ1033330-1-11+5/21-5/20001-5/21+5/21+5/21-5/21+5/21-5/2    orthogonal lifted from A5
ρ114-44-4100-1-11-1-11111-1-1    orthogonal lifted from C2×A5
ρ124444100-1-1111-1-1-1-1-1-1    orthogonal lifted from A5
ρ1344-4-4100-1-1-1-11-1-11111    symplectic lifted from SL2(𝔽5), Schur index 2
ρ144-4-44100-1-1-11-111-1-111    symplectic lifted from SL2(𝔽5), Schur index 2
ρ155-55-5-11-100-111000000    orthogonal lifted from C2×A5
ρ165555-11100-1-1-1000000    orthogonal lifted from A5
ρ1766-6-60001100011-1-1-1-1    symplectic lifted from SL2(𝔽5), Schur index 2
ρ186-6-6600011000-1-111-1-1    symplectic lifted from SL2(𝔽5), Schur index 2

Smallest permutation representation of C2×SL2(𝔽5)
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 26 40 47 25 3 12 30 37 11)(2 21 45 42 20 4 17 35 32 16)(5 29 22 19 38 7 39 18 15 48)(6 34 27 24 33 8 44 13 10 43)(9 23)(14 28)(31 41)(36 46)

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

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

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

C2×SL2(𝔽5) is a maximal subgroup of   C22.2S5  C22.S5  D4.A5

Matrix representation of C2×SL2(𝔽5) in GL3(𝔽61) generated by

6000
0722
05336
,
6000
02030
01824
G:=sub<GL(3,GF(61))| [60,0,0,0,7,53,0,22,36],[60,0,0,0,20,18,0,30,24] >;

C2×SL2(𝔽5) in GAP, Magma, Sage, TeX

C_2\times {\rm SL}_2({\mathbb F}_5)
% in TeX

G:=Group("C2xSL(2,5)");
// GroupNames label

G:=SmallGroup(240,94);
// by ID

G=gap.SmallGroup(240,94);
# by ID

Export

Subgroup lattice of C2×SL2(𝔽5) in TeX
Character table of C2×SL2(𝔽5) in TeX

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