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## G = C2×D5×S4order 480 = 25·3·5

### Direct product of C2, D5 and S4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×A4 — C2×D5×S4
 Chief series C1 — C22 — C2×C10 — C5×A4 — D5×A4 — D5×S4 — C2×D5×S4
 Lower central C5×A4 — C2×D5×S4
 Upper central C1 — C2

Generators and relations for C2×D5×S4
G = < a,b,c,d,e,f,g | a2=b5=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, cbc=b-1, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 2128 in 262 conjugacy classes, 35 normal (19 characteristic)
C1, C2, C2, C3, C4, C22, C22, C5, S3, C6, C2×C4, D4, C23, C23, D5, D5, C10, C10, A4, D6, C2×C6, C15, C22×C4, C2×D4, C24, Dic5, C20, D10, D10, C2×C10, C2×C10, S4, S4, C2×A4, C2×A4, C22×S3, C5×S3, C3×D5, D15, C30, C22×D4, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C22×D5, C22×D5, C22×C10, C22×C10, C2×S4, C2×S4, C22×A4, S3×D5, C5×A4, C6×D5, S3×C10, D30, C2×C4×D5, C2×D20, D4×D5, C2×C5⋊D4, D4×C10, C23×D5, C23×D5, C22×S4, C5×S4, C5⋊S4, D5×A4, C2×S3×D5, C10×A4, C2×D4×D5, D5×S4, C10×S4, C2×C5⋊S4, C2×D5×A4, C2×D5×S4
Quotients: C1, C2, C22, S3, C23, D5, D6, D10, S4, C22×S3, C22×D5, C2×S4, S3×D5, C22×S4, C2×S3×D5, D5×S4, C2×D5×S4

Permutation representations of C2×D5×S4
On 30 points - transitive group 30T109
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 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 5)(2 4)(6 9)(7 8)(11 14)(12 13)(16 17)(18 20)(21 24)(22 23)(26 27)(28 30)
(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)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

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

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

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

G:=TransitiveGroup(30,109);

40 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 3 4A 4B 4C 4D 5A 5B 6A 6B 6C 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 15A 15B 20A 20B 20C 20D 30A 30B order 1 2 2 2 2 2 2 2 2 2 2 2 3 4 4 4 4 5 5 6 6 6 10 10 10 10 10 10 10 10 10 10 15 15 20 20 20 20 30 30 size 1 1 3 3 5 5 6 6 15 15 30 30 8 6 6 30 30 2 2 8 40 40 2 2 6 6 6 6 12 12 12 12 16 16 12 12 12 12 16 16

40 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 3 3 3 4 4 6 6 type + + + + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 S3 D5 D6 D6 D10 D10 S4 C2×S4 C2×S4 S3×D5 C2×S3×D5 D5×S4 C2×D5×S4 kernel C2×D5×S4 D5×S4 C10×S4 C2×C5⋊S4 C2×D5×A4 C23×D5 C2×S4 C22×D5 C22×C10 S4 C2×A4 D10 D5 C10 C23 C22 C2 C1 # reps 1 4 1 1 1 1 2 2 1 4 2 2 4 2 2 2 4 4

Matrix representation of C2×D5×S4 in GL5(𝔽61)

 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 60
,
 0 1 0 0 0 60 17 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 0 60 0 0 0 60 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 60 0 0 0 0 0 60
,
 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 0
,
 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0

G:=sub<GL(5,GF(61))| [60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,60],[0,60,0,0,0,1,17,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,60,0,0,0,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60,0,0,0,0,0,1],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,60,0,0,0,0,0,60],[1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0,0],[1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0] >;

C2×D5×S4 in GAP, Magma, Sage, TeX

C_2\times D_5\times S_4
% in TeX

G:=Group("C2xD5xS4");
// GroupNames label

G:=SmallGroup(480,1193);
// by ID

G=gap.SmallGroup(480,1193);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,234,3364,5052,1286,2953,2232]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^2=b^5=c^2=d^2=e^2=f^3=g^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*d*f^-1=g*d*g=d*e=e*d,f*e*f^-1=d,e*g=g*e,g*f*g=f^-1>;
// generators/relations

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