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
| C5×A4 — C2×A4⋊F5 |
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 |
►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
Export