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## G = C2×A4⋊F5order 480 = 25·3·5

### Direct product of C2 and A4⋊F5

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
 Upper central C1 — C2

Generators and relations for C2×A4⋊F5
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 >

Subgroups: 1024 in 126 conjugacy classes, 24 normal (22 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, D5, D5, C10, C10, Dic3, A4, C2×C6, C15, C22⋊C4, C22×C4, C24, F5, D10, D10, C2×C10, C2×C10, C2×Dic3, C2×A4, C2×A4, C3×D5, C30, C2×C22⋊C4, C2×F5, C22×D5, C22×D5, C22×C10, A4⋊C4, C22×A4, C3⋊F5, C5×A4, C6×D5, C22⋊F5, C22×F5, C23×D5, C2×A4⋊C4, D5×A4, C2×C3⋊F5, C10×A4, C2×C22⋊F5, A4⋊F5, C2×D5×A4, C2×A4⋊F5
Quotients: C1, C2, C4, C22, S3, C2×C4, Dic3, D6, F5, C2×Dic3, S4, C2×F5, A4⋊C4, C2×S4, C3⋊F5, C2×A4⋊C4, C2×C3⋊F5, A4⋊F5, C2×A4⋊F5

Character table of C2×A4⋊F5

 class 1 2A 2B 2C 2D 2E 2F 2G 3 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 10A 10B 10C 15A 15B 30A 30B size 1 1 3 3 5 5 15 15 8 30 30 30 30 30 30 30 30 4 8 40 40 4 12 12 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 -1 1 -1 -1 1 1 1 -1 -1 -1 1 1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 1 1 1 1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 -1 linear of order 2 ρ5 1 -1 1 -1 -1 1 1 -1 1 -i -i i i i i -i -i 1 -1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 4 ρ6 1 1 1 1 -1 -1 -1 -1 1 -i i -i -i i i -i i 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 4 ρ7 1 -1 1 -1 -1 1 1 -1 1 i i -i -i -i -i i i 1 -1 -1 1 -1 1 -1 1 1 -1 -1 linear of order 4 ρ8 1 1 1 1 -1 -1 -1 -1 1 i -i i i -i -i i -i 1 1 -1 -1 1 1 1 1 1 1 1 linear of order 4 ρ9 2 2 2 2 2 2 2 2 -1 0 0 0 0 0 0 0 0 2 -1 -1 -1 2 2 2 -1 -1 -1 -1 orthogonal lifted from S3 ρ10 2 -2 2 -2 2 -2 -2 2 -1 0 0 0 0 0 0 0 0 2 1 -1 1 -2 2 -2 -1 -1 1 1 orthogonal lifted from D6 ρ11 2 2 2 2 -2 -2 -2 -2 -1 0 0 0 0 0 0 0 0 2 -1 1 1 2 2 2 -1 -1 -1 -1 symplectic lifted from Dic3, Schur index 2 ρ12 2 -2 2 -2 -2 2 2 -2 -1 0 0 0 0 0 0 0 0 2 1 1 -1 -2 2 -2 -1 -1 1 1 symplectic lifted from Dic3, Schur index 2 ρ13 3 3 -1 -1 3 3 -1 -1 0 1 1 -1 1 -1 1 -1 -1 3 0 0 0 3 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ14 3 -3 -1 1 3 -3 1 -1 0 -1 1 -1 1 1 -1 1 -1 3 0 0 0 -3 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ15 3 3 -1 -1 3 3 -1 -1 0 -1 -1 1 -1 1 -1 1 1 3 0 0 0 3 -1 -1 0 0 0 0 orthogonal lifted from S4 ρ16 3 -3 -1 1 3 -3 1 -1 0 1 -1 1 -1 -1 1 -1 1 3 0 0 0 -3 -1 1 0 0 0 0 orthogonal lifted from C2×S4 ρ17 3 -3 -1 1 -3 3 -1 1 0 i i i -i i -i -i -i 3 0 0 0 -3 -1 1 0 0 0 0 complex lifted from A4⋊C4 ρ18 3 3 -1 -1 -3 -3 1 1 0 -i i i -i -i i i -i 3 0 0 0 3 -1 -1 0 0 0 0 complex lifted from A4⋊C4 ρ19 3 -3 -1 1 -3 3 -1 1 0 -i -i -i i -i i i i 3 0 0 0 -3 -1 1 0 0 0 0 complex lifted from A4⋊C4 ρ20 3 3 -1 -1 -3 -3 1 1 0 i -i -i i i -i -i i 3 0 0 0 3 -1 -1 0 0 0 0 complex lifted from A4⋊C4 ρ21 4 -4 4 -4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 -4 0 0 1 -1 1 -1 -1 1 1 orthogonal lifted from C2×F5 ρ22 4 4 4 4 0 0 0 0 4 0 0 0 0 0 0 0 0 -1 4 0 0 -1 -1 -1 -1 -1 -1 -1 orthogonal lifted from F5 ρ23 4 4 4 4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -1 -2 0 0 -1 -1 -1 1+√-15/2 1-√-15/2 1-√-15/2 1+√-15/2 complex lifted from C3⋊F5 ρ24 4 4 4 4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -1 -2 0 0 -1 -1 -1 1-√-15/2 1+√-15/2 1+√-15/2 1-√-15/2 complex lifted from C3⋊F5 ρ25 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -1 2 0 0 1 -1 1 1+√-15/2 1-√-15/2 -1+√-15/2 -1-√-15/2 complex lifted from C2×C3⋊F5 ρ26 4 -4 4 -4 0 0 0 0 -2 0 0 0 0 0 0 0 0 -1 2 0 0 1 -1 1 1-√-15/2 1+√-15/2 -1-√-15/2 -1+√-15/2 complex lifted from C2×C3⋊F5 ρ27 12 -12 -4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 0 0 0 3 1 -1 0 0 0 0 orthogonal faithful ρ28 12 12 -4 -4 0 0 0 0 0 0 0 0 0 0 0 0 0 -3 0 0 0 -3 1 1 0 0 0 0 orthogonal lifted from A4⋊F5

Permutation representations of C2×A4⋊F5
On 30 points - transitive group 30T121
Generators in S30
(1 11)(2 12)(3 13)(4 14)(5 15)(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 11)(2 12)(3 13)(4 14)(5 15)(16 25)(17 21)(18 22)(19 23)(20 24)
(6 30)(7 26)(8 27)(9 28)(10 29)(16 25)(17 21)(18 22)(19 23)(20 24)
(1 17 27)(2 18 28)(3 19 29)(4 20 30)(5 16 26)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(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 22 10 25)(7 24 9 23)(8 21)(12 13 15 14)(16 30 18 29)(17 27)(19 26 20 28)

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

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

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

G:=TransitiveGroup(30,121);

Matrix representation of C2×A4⋊F5 in GL7(𝔽61)

 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 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 27 0 55 55 0 0 0 6 33 6 0 0 0 0 0 6 33 6 0 0 0 55 55 0 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 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 60 60 60 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 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 60 60 60 60

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] >;

C2×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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