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G = C2×A4⋊F5  order 480 = 25·3·5

Direct product of C2 and A4⋊F5

direct product, non-abelian, soluble, monomial

Aliases: C2×A4⋊F5, D10.4S4, (C2×A4)⋊F5, C10⋊(A4⋊C4), D5⋊(A4⋊C4), C23⋊(C3⋊F5), (D5×A4)⋊2C4, A4⋊2(C2×F5), (C10×A4)⋊1C4, D5.2(C2×S4), (C22×C10)⋊Dic3, (C23×D5).2S3, (D5×A4).2C22, (C22×D5).2D6, (C22×D5)⋊2Dic3, C5⋊(C2×A4⋊C4), C22⋊(C2×C3⋊F5), (C2×D5×A4).2C2, (C5×A4)⋊2(C2×C4), (C2×C10)⋊1(C2×Dic3), SmallGroup(480,1191)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C22 — C5×A4 — C2×A4⋊F5
Chief series
C1 — C22 — C2×C10 — C5×A4 — D5×A4 — A4⋊F5 — C2×A4⋊F5
Lower central
C5×A4 — C2×A4⋊F5

Subgroups: 1024 in 126 conjugacy classes, 24 normal (22 characteristic)
C1, C2, C2 [×6], C3, C4 [×4], C22, C22 [×11], C5, C6 [×3], C2×C4 [×8], C23, C23 [×6], D5 [×2], D5 [×2], C10, C10 [×2], Dic3 [×2], A4, C2×C6, C15, C22⋊C4 [×4], C22×C4 [×2], C24, F5 [×4], D10, D10 [×8], C2×C10, C2×C10 [×2], C2×Dic3, C2×A4, C2×A4 [×2], C3×D5 [×2], C30, C2×C22⋊C4, C2×F5 [×8], C22×D5 [×2], C22×D5 [×4], C22×C10, A4⋊C4 [×2], C22×A4, C3⋊F5 [×2], C5×A4, C6×D5, C22⋊F5 [×4], C22×F5 [×2], C23×D5, C2×A4⋊C4, D5×A4 [×2], C2×C3⋊F5, C10×A4, C2×C22⋊F5, A4⋊F5 [×2], C2×D5×A4, C2×A4⋊F5

Quotients:
C1, C2 [×3], C4 [×2], C22, S3, C2×C4, Dic3 [×2], D6, F5, C2×Dic3, S4, C2×F5, A4⋊C4 [×2], C2×S4, C3⋊F5, C2×A4⋊C4, C2×C3⋊F5, A4⋊F5, C2×A4⋊F5

Generators and relations
 G = < a,b,c,d,e,f | a2=b2=c2=d3=e5=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, dbd-1=fbf-1=bc=cb, be=eb, dcd-1=b, ce=ec, cf=fc, de=ed, fdf-1=d-1, fef-1=e3 >

Permutation representations
►On 30 points - transitive group 30T121
Generators in S30
(1 13)(2 14)(3 15)(4 11)(5 12)(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

(1 13)(2 14)(3 15)(4 11)(5 12)(16 22)(17 23)(18 24)(19 25)(20 21)

(6 30)(7 26)(8 27)(9 28)(10 29)(16 22)(17 23)(18 24)(19 25)(20 21)

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

(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)

(2 3 5 4)(6 24 10 22)(7 21 9 25)(8 23)(11 14 15 12)(16 30 18 29)(17 27)(19 26 20 28)
 
G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (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), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,30,18,29)(17,27)(19,26,20,28)>;
 
G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,13)(2,14)(3,15)(4,11)(5,12)(16,22)(17,23)(18,24)(19,25)(20,21), (6,30)(7,26)(8,27)(9,28)(10,29)(16,22)(17,23)(18,24)(19,25)(20,21), (1,17,27)(2,18,28)(3,19,29)(4,20,30)(5,16,26)(6,11,21)(7,12,22)(8,13,23)(9,14,24)(10,15,25), (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), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,30,18,29)(17,27)(19,26,20,28) );
 
G=PermutationGroup([(1,13),(2,14),(3,15),(4,11),(5,12),(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,13),(2,14),(3,15),(4,11),(5,12),(16,22),(17,23),(18,24),(19,25),(20,21)], [(6,30),(7,26),(8,27),(9,28),(10,29),(16,22),(17,23),(18,24),(19,25),(20,21)], [(1,17,27),(2,18,28),(3,19,29),(4,20,30),(5,16,26),(6,11,21),(7,12,22),(8,13,23),(9,14,24),(10,15,25)], [(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)], [(2,3,5,4),(6,24,10,22),(7,21,9,25),(8,23),(11,14,15,12),(16,30,18,29),(17,27),(19,26,20,28)])
 
G:=TransitiveGroup(30,121);
 

Matrix representation ►G ⊆ GL7(𝔽61)

60000000
06000000
00600000
0001000
0000100
0000010
0000001
,
60000000
06000000
0010000
0001000
0000100
0000010
0000001
,
1000000
06000000
00600000
0001000
0000100
0000010
0000001
,
0010000
1000000
0100000
0002705555
00063360
00006336
0005555027
,
1000000
0100000
0010000
0000100
0000010
0000001
00060606060
,
50000000
00500000
05000000
0001000
0000001
0000100
00060606060

G:=sub<GL(7,GF(61))| [60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,27,6,0,55,0,0,0,0,33,6,55,0,0,0,55,6,33,0,0,0,0,55,0,6,27],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,60,0,0,0,1,0,0,60,0,0,0,0,1,0,60,0,0,0,0,0,1,60],[50,0,0,0,0,0,0,0,0,50,0,0,0,0,0,50,0,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;
 

Character table of C2×A4⋊F5

 class 12A2B2C2D2E2F2G34A4B4C4D4E4F4G4H56A6B6C10A10B10C15A15B30A30B
 size 1133551515830303030303030304840404121216161616
ρ11111111111111111111111111111    trivial
ρ21-11-11-1-1111-1-1-1111-11-11-1-11-111-1-1    linear of order 2
ρ3111111111-1-1-1-1-1-1-1-111111111111    linear of order 2
ρ41-11-11-1-111-1111-1-1-111-11-1-11-111-1-1    linear of order 2
ρ51-11-1-111-11-i-iiiii-i-i1-1-11-11-111-1-1    linear of order 4
ρ61111-1-1-1-11-ii-i-iii-ii11-1-11111111    linear of order 4
ρ71-11-1-111-11ii-i-i-i-iii1-1-11-11-111-1-1    linear of order 4
ρ81111-1-1-1-11i-iii-i-ii-i11-1-11111111    linear of order 4
ρ922222222-1000000002-1-1-1222-1-1-1-1    orthogonal lifted from S3
ρ102-22-22-2-22-10000000021-11-22-2-1-111    orthogonal lifted from D6
ρ112222-2-2-2-2-1000000002-111222-1-1-1-1    symplectic lifted from Dic3, Schur index 2
ρ122-22-2-222-2-100000000211-1-22-2-1-111    symplectic lifted from Dic3, Schur index 2
ρ1333-1-133-1-1011-11-11-1-130003-1-10000    orthogonal lifted from S4
ρ143-3-113-31-10-11-111-11-13000-3-110000    orthogonal lifted from C2×S4
ρ1533-1-133-1-10-1-11-11-11130003-1-10000    orthogonal lifted from S4
ρ163-3-113-31-101-11-1-11-113000-3-110000    orthogonal lifted from C2×S4
ρ173-3-11-33-110iii-ii-i-i-i3000-3-110000    complex lifted from A4⋊C4
ρ1833-1-1-3-3110-iii-i-iii-i30003-1-10000    complex lifted from A4⋊C4
ρ193-3-11-33-110-i-i-ii-iiii3000-3-110000    complex lifted from A4⋊C4
ρ2033-1-1-3-3110i-i-iii-i-ii30003-1-10000    complex lifted from A4⋊C4
ρ214-44-40000400000000-1-4001-11-1-111    orthogonal lifted from C2×F5
ρ2244440000400000000-1400-1-1-1-1-1-1-1    orthogonal lifted from F5
ρ2344440000-200000000-1-200-1-1-11+√-15/21-√-15/21-√-15/21+√-15/2    complex lifted from C3⋊F5
ρ2444440000-200000000-1-200-1-1-11-√-15/21+√-15/21+√-15/21-√-15/2    complex lifted from C3⋊F5
ρ254-44-40000-200000000-12001-111+√-15/21-√-15/2-1+√-15/2-1-√-15/2    complex lifted from C2×C3⋊F5
ρ264-44-40000-200000000-12001-111-√-15/21+√-15/2-1-√-15/2-1+√-15/2    complex lifted from C2×C3⋊F5
ρ2712-12-440000000000000-300031-10000    orthogonal faithful
ρ281212-4-40000000000000-3000-3110000    orthogonal lifted from A4⋊F5

In GAP, Magma, Sage, TeX 

C_2\times A_4\rtimes F_5
 
% in TeX
 
G:=Group("C2xA4:F5");
 
// GroupNames label
 
G:=SmallGroup(480,1191);
 
// by ID
 
G=gap.SmallGroup(480,1191);
 
# by ID
 
G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,451,2524,858,10085,1286,5886,2232]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^3=e^5=f^4=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,d*b*d^-1=f*b*f^-1=b*c=c*b,b*e=e*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,f*d*f^-1=d^-1,f*e*f^-1=e^3>;
 
// generators/relations
 

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