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G = C5×A5order 300 = 22·3·52

Direct product of C5 and A5

direct product, non-abelian, not soluble, A-group

Aliases: C5×A5, U2(𝔽4), SmallGroup(300,22)

Series: ChiefDerived Lower central Upper central

C1C5 — C5×A5
A5 — C5×A5
A5 — C5×A5
C1C5

15C2
10C3
6C5
12C5
12C5
5C22
10S3
6D5
15C10
10C15
6C52
5A4
5C2×C10
10C5×S3
6C5×D5
5C5×A4

Character table of C5×A5

 class 1235A5B5C5D5E5F5G5H5I5J5K5L5M5N10A10B10C10D15A15B15C15D
 size 115201111121212121212121212121515151520202020
ρ11111111111111111111111111    trivial
ρ2111ζ54ζ52ζ53ζ5ζ54ζ53ζ52ζ511ζ54ζ53ζ52ζ5ζ5ζ53ζ52ζ54ζ53ζ5ζ52ζ54    linear of order 5
ρ3111ζ52ζ5ζ54ζ53ζ52ζ54ζ5ζ5311ζ52ζ54ζ5ζ53ζ53ζ54ζ5ζ52ζ54ζ53ζ5ζ52    linear of order 5
ρ4111ζ53ζ54ζ5ζ52ζ53ζ5ζ54ζ5211ζ53ζ5ζ54ζ52ζ52ζ5ζ54ζ53ζ5ζ52ζ54ζ53    linear of order 5
ρ5111ζ5ζ53ζ52ζ54ζ5ζ52ζ53ζ5411ζ5ζ52ζ53ζ54ζ54ζ52ζ53ζ5ζ52ζ54ζ53ζ5    linear of order 5
ρ63-1033331-5/21+5/21+5/21-5/21+5/21-5/21+5/21-5/21-5/21+5/2-1-1-1-10000    orthogonal lifted from A5
ρ73-1033331+5/21-5/21-5/21+5/21-5/21+5/21-5/21+5/21+5/21-5/2-1-1-1-10000    orthogonal lifted from A5
ρ83-105535254545353554525251-5/21+5/252-154-15-153-154525350000    complex faithful
ρ93-105255453535525545354521+5/21-5/254-153-152-15-153545520000    complex faithful
ρ103-105452535525545253554531-5/21+5/253-15-154-152-155352540000    complex faithful
ρ113-10545253553-15-154-152-11+5/21-5/25255452535545355352540000    complex faithful
ρ123-10525545354-153-152-15-11-5/21+5/25355255453545253545520000    complex faithful
ρ133-10553525452-154-15-153-11+5/21-5/25453535545252554525350000    complex faithful
ρ143-105354552545254535255351+5/21-5/25-152-153-154-152554530000    complex faithful
ρ153-1053545525-152-153-154-11-5/21+5/25452545352553552554530000    complex faithful
ρ164014444-1-1-1-1-1-1-1-1-1-100001111    orthogonal lifted from A5
ρ1740152554535254553-1-152545530000ζ54ζ53ζ5ζ52    complex faithful
ρ1840155352545525354-1-155253540000ζ52ζ54ζ53ζ5    complex faithful
ρ1940154525355453525-1-154535250000ζ53ζ5ζ52ζ54    complex faithful
ρ2040153545525355452-1-153554520000ζ5ζ52ζ54ζ53    complex faithful
ρ2151-1555500000000001111-1-1-1-1    orthogonal lifted from A5
ρ2251-152554530000000000ζ53ζ54ζ5ζ525453552    complex faithful
ρ2351-154525350000000000ζ5ζ53ζ52ζ545355254    complex faithful
ρ2451-153545520000000000ζ52ζ5ζ54ζ535525453    complex faithful
ρ2551-155352540000000000ζ54ζ52ζ53ζ55254535    complex faithful

Permutation representations of C5×A5
On 25 points - transitive group 25T29
Generators in S25
(1 7 14 12 25)(2 8 20 18 16)(3 9 11 24 22)(4 10 17 15 13)(5 6 23 21 19)
(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)

G:=sub<Sym(25)| (1,7,14,12,25)(2,8,20,18,16)(3,9,11,24,22)(4,10,17,15,13)(5,6,23,21,19), (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)>;

G:=Group( (1,7,14,12,25)(2,8,20,18,16)(3,9,11,24,22)(4,10,17,15,13)(5,6,23,21,19), (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) );

G=PermutationGroup([(1,7,14,12,25),(2,8,20,18,16),(3,9,11,24,22),(4,10,17,15,13),(5,6,23,21,19)], [(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)])

G:=TransitiveGroup(25,29);

On 30 points - transitive group 30T69
Generators in S30
(1 9 20 5 21)(2 18 13 6 17)(3 29 14 30 10)(4 12 23 8 24)(7 15 26 11 27)(16 19 22 25 28)
(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)

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

G:=Group( (1,9,20,5,21)(2,18,13,6,17)(3,29,14,30,10)(4,12,23,8,24)(7,15,26,11,27)(16,19,22,25,28), (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) );

G=PermutationGroup([(1,9,20,5,21),(2,18,13,6,17),(3,29,14,30,10),(4,12,23,8,24),(7,15,26,11,27),(16,19,22,25,28)], [(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)])

G:=TransitiveGroup(30,69);

Matrix representation of C5×A5 in GL3(𝔽11) generated by

1090
330
381
,
069
014
5510
G:=sub<GL(3,GF(11))| [10,3,3,9,3,8,0,0,1],[0,0,5,6,1,5,9,4,10] >;

C5×A5 in GAP, Magma, Sage, TeX

C_5\times A_5
% in TeX

G:=Group("C5xA5");
// GroupNames label

G:=SmallGroup(300,22);
// by ID

G=gap.SmallGroup(300,22);
# by ID

Export

Subgroup lattice of C5×A5 in TeX
Character table of C5×A5 in TeX

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