Copied to
clipboard

## G = C2×A4×F5order 480 = 25·3·5

### Direct product of C2, A4 and F5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — C2×A4×F5
 Chief series C1 — C5 — C2×C10 — C22×D5 — D5×A4 — A4×F5 — C2×A4×F5
 Lower central C2×C10 — 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=bc=cb, be=eb, bf=fb, dcd-1=b, ce=ec, cf=fc, de=ed, df=fd, fef-1=e3 >

Subgroups: 844 in 132 conjugacy classes, 30 normal (24 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, C6, C2×C4, C23, C23, D5, D5, C10, C10, C12, A4, C2×C6, C15, C22×C4, C24, F5, F5, D10, D10, C2×C10, C2×C10, C2×C12, C2×A4, C2×A4, C3×D5, C30, C23×C4, C2×F5, C2×F5, C22×D5, C22×D5, C22×C10, C4×A4, C22×A4, C3×F5, C5×A4, C6×D5, C22×F5, C22×F5, C23×D5, C2×C4×A4, D5×A4, C6×F5, C10×A4, C23×F5, A4×F5, C2×D5×A4, C2×A4×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, A4, C2×C6, F5, C2×C12, C2×A4, C2×F5, C4×A4, C22×A4, C3×F5, C2×C4×A4, C6×F5, A4×F5, C2×A4×F5

Permutation representations of C2×A4×F5
On 30 points - transitive group 30T107
Generators in S30
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)
(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)
(1 6)(2 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)(21 26)(22 28 25 29)(23 30 24 27)

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

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

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

G:=TransitiveGroup(30,107);

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 3A 3B 4A 4B 4C 4D 4E 4F 4G 4H 5 6A 6B 6C 6D 6E 6F 10A 10B 10C 12A ··· 12H 15A 15B 30A 30B order 1 2 2 2 2 2 2 2 3 3 4 4 4 4 4 4 4 4 5 6 6 6 6 6 6 10 10 10 12 ··· 12 15 15 30 30 size 1 1 3 3 5 5 15 15 4 4 5 5 5 5 15 15 15 15 4 4 4 20 20 20 20 4 12 12 20 ··· 20 16 16 16 16

40 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 12 12 3 3 3 3 3 4 4 4 4 type + + + + + + + + + + image C1 C2 C2 C3 C4 C4 C6 C6 C12 C12 A4×F5 C2×A4×F5 A4 C2×A4 C2×A4 C4×A4 C4×A4 F5 C2×F5 C3×F5 C6×F5 kernel C2×A4×F5 A4×F5 C2×D5×A4 C23×F5 D5×A4 C10×A4 C22×F5 C23×D5 C22×D5 C22×C10 C2 C1 C2×F5 F5 D10 D5 C10 C2×A4 A4 C23 C22 # reps 1 2 1 2 2 2 4 2 4 4 1 1 1 2 1 2 2 1 1 2 2

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 14 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
,
 60 0 0 0 0 0 0 48 1 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
,
 13 59 0 0 0 0 0 0 48 1 0 0 0 0 0 14 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 0 0 0 0 1
,
 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 60 60 60 60 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 50 0 0 0 0 0 0 0 50 0 0 0 0 0 0 0 50 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],[60,0,14,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[60,48,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[13,0,0,0,0,0,0,59,48,14,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,1,0,0,0,60,0,0,0],[50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;

C2×A4×F5 in GAP, Magma, Sage, TeX

C_2\times A_4\times F_5
% in TeX

G:=Group("C2xA4xF5");
// GroupNames label

G:=SmallGroup(480,1192);
// by ID

G=gap.SmallGroup(480,1192);
# by ID

G:=PCGroup([7,-2,-2,-3,-2,-2,2,-5,84,648,271,9414,1595]);
// 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=b*c=c*b,b*e=e*b,b*f=f*b,d*c*d^-1=b,c*e=e*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f^-1=e^3>;
// generators/relations

׿
×
𝔽