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## G = C5×C22⋊S4order 480 = 25·3·5

### Direct product of C5 and C22⋊S4

Aliases: C5×C22⋊S4, C22⋊(C5×S4), (C2×C10)⋊1S4, C243(C5×S3), C22⋊A43C10, (C23×C10)⋊3S3, (C5×C22⋊A4)⋊5C2, SmallGroup(480,1200)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22⋊A4 — C5×C22⋊S4
 Chief series C1 — C22 — C24 — C22⋊A4 — C5×C22⋊A4 — C5×C22⋊S4
 Lower central C22⋊A4 — C5×C22⋊S4
 Upper central C1 — C5

Generators and relations for C5×C22⋊S4
G = < a,b,c,d,e,f,g | a5=b2=c2=d2=e2=f3=g2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, fcf-1=bc=cb, bd=db, be=eb, fbf-1=gbg=c, cd=dc, ce=ec, gcg=b, fdf-1=gdg=de=ed, fef-1=d, eg=ge, gfg=f-1 >

Subgroups: 500 in 112 conjugacy classes, 14 normal (8 characteristic)
C1, C2, C3, C4, C22, C22, C5, S3, C2×C4, D4, C23, C10, A4, C15, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, S4, C5×S3, C22≀C2, C2×C20, C5×D4, C22×C10, C22⋊A4, C5×A4, C5×C22⋊C4, D4×C10, C23×C10, C22⋊S4, C5×S4, C5×C22≀C2, C5×C22⋊A4, C5×C22⋊S4
Quotients: C1, C2, C5, S3, C10, S4, C5×S3, C22⋊S4, C5×S4, C5×C22⋊S4

Smallest permutation representation of C5×C22⋊S4
On 40 points
Generators in S40
(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)(31 32 33 34 35)(36 37 38 39 40)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 31)(25 32)
(1 23)(2 24)(3 25)(4 21)(5 22)(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)(21 28)(22 29)(23 30)(24 26)(25 27)
(1 23)(2 24)(3 25)(4 21)(5 22)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(6 16 11)(7 17 12)(8 18 13)(9 19 14)(10 20 15)(21 28 33)(22 29 34)(23 30 35)(24 26 31)(25 27 32)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35)(12 31)(13 32)(14 33)(15 34)(16 23)(17 24)(18 25)(19 21)(20 22)

G:=sub<Sym(40)| (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)(31,32,33,34,35)(36,37,38,39,40), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,31)(25,32), (1,23)(2,24)(3,25)(4,21)(5,22)(6,36)(7,37)(8,38)(9,39)(10,40)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35), (1,35)(2,31)(3,32)(4,33)(5,34)(6,36)(7,37)(8,38)(9,39)(10,40)(11,16)(12,17)(13,18)(14,19)(15,20)(21,28)(22,29)(23,30)(24,26)(25,27), (1,23)(2,24)(3,25)(4,21)(5,22)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(26,31)(27,32)(28,33)(29,34)(30,35), (6,16,11)(7,17,12)(8,18,13)(9,19,14)(10,20,15)(21,28,33)(22,29,34)(23,30,35)(24,26,31)(25,27,32), (1,36)(2,37)(3,38)(4,39)(5,40)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22)>;

G:=Group( (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)(31,32,33,34,35)(36,37,38,39,40), (1,30)(2,26)(3,27)(4,28)(5,29)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(21,33)(22,34)(23,35)(24,31)(25,32), (1,23)(2,24)(3,25)(4,21)(5,22)(6,36)(7,37)(8,38)(9,39)(10,40)(11,16)(12,17)(13,18)(14,19)(15,20)(26,31)(27,32)(28,33)(29,34)(30,35), (1,35)(2,31)(3,32)(4,33)(5,34)(6,36)(7,37)(8,38)(9,39)(10,40)(11,16)(12,17)(13,18)(14,19)(15,20)(21,28)(22,29)(23,30)(24,26)(25,27), (1,23)(2,24)(3,25)(4,21)(5,22)(6,11)(7,12)(8,13)(9,14)(10,15)(16,36)(17,37)(18,38)(19,39)(20,40)(26,31)(27,32)(28,33)(29,34)(30,35), (6,16,11)(7,17,12)(8,18,13)(9,19,14)(10,20,15)(21,28,33)(22,29,34)(23,30,35)(24,26,31)(25,27,32), (1,36)(2,37)(3,38)(4,39)(5,40)(6,30)(7,26)(8,27)(9,28)(10,29)(11,35)(12,31)(13,32)(14,33)(15,34)(16,23)(17,24)(18,25)(19,21)(20,22) );

G=PermutationGroup([[(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),(31,32,33,34,35),(36,37,38,39,40)], [(1,30),(2,26),(3,27),(4,28),(5,29),(6,11),(7,12),(8,13),(9,14),(10,15),(16,36),(17,37),(18,38),(19,39),(20,40),(21,33),(22,34),(23,35),(24,31),(25,32)], [(1,23),(2,24),(3,25),(4,21),(5,22),(6,36),(7,37),(8,38),(9,39),(10,40),(11,16),(12,17),(13,18),(14,19),(15,20),(26,31),(27,32),(28,33),(29,34),(30,35)], [(1,35),(2,31),(3,32),(4,33),(5,34),(6,36),(7,37),(8,38),(9,39),(10,40),(11,16),(12,17),(13,18),(14,19),(15,20),(21,28),(22,29),(23,30),(24,26),(25,27)], [(1,23),(2,24),(3,25),(4,21),(5,22),(6,11),(7,12),(8,13),(9,14),(10,15),(16,36),(17,37),(18,38),(19,39),(20,40),(26,31),(27,32),(28,33),(29,34),(30,35)], [(6,16,11),(7,17,12),(8,18,13),(9,19,14),(10,20,15),(21,28,33),(22,29,34),(23,30,35),(24,26,31),(25,27,32)], [(1,36),(2,37),(3,38),(4,39),(5,40),(6,30),(7,26),(8,27),(9,28),(10,29),(11,35),(12,31),(13,32),(14,33),(15,34),(16,23),(17,24),(18,25),(19,21),(20,22)]])

50 conjugacy classes

 class 1 2A 2B 2C 2D 2E 3 4A 4B 4C 5A 5B 5C 5D 10A ··· 10L 10M 10N 10O 10P 10Q 10R 10S 10T 15A 15B 15C 15D 20A ··· 20L order 1 2 2 2 2 2 3 4 4 4 5 5 5 5 10 ··· 10 10 10 10 10 10 10 10 10 15 15 15 15 20 ··· 20 size 1 3 3 3 6 12 32 12 12 12 1 1 1 1 3 ··· 3 6 6 6 6 12 12 12 12 32 32 32 32 12 ··· 12

50 irreducible representations

 dim 1 1 1 1 2 2 3 3 6 6 type + + + + + image C1 C2 C5 C10 S3 C5×S3 S4 C5×S4 C22⋊S4 C5×C22⋊S4 kernel C5×C22⋊S4 C5×C22⋊A4 C22⋊S4 C22⋊A4 C23×C10 C24 C2×C10 C22 C5 C1 # reps 1 1 4 4 1 4 6 24 1 4

Matrix representation of C5×C22⋊S4 in GL6(𝔽61)

 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 0 0 0 0 0 0 34 0 0 0 0 0 0 34
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 60 60 60 0 0 0 1 0 0
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 60 60 60 0 0 0 0 0 1 0 0 0 0 1 0
,
 60 0 0 0 0 0 60 0 1 0 0 0 60 1 0 0 0 0 0 0 0 0 0 1 0 0 0 60 60 60 0 0 0 1 0 0
,
 0 1 60 0 0 0 1 0 60 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 60 60 60
,
 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 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 60 0 0 0 0 60 0 0 0 0 0 0 0 60

G:=sub<GL(6,GF(61))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,0,0,0,0,0,0,34,0,0,0,0,0,0,34],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,1,60,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,60,0,0,0,0,0,60,0,1,0,0,0,60,1,0],[60,60,60,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,60,1,0,0,0,0,60,0,0,0,0,1,60,0],[0,1,0,0,0,0,1,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,60,0,0,0,1,0,60,0,0,0,0,0,60],[0,0,1,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,1,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,60,0,0,0,0,0,0,0,60] >;

C5×C22⋊S4 in GAP, Magma, Sage, TeX

C_5\times C_2^2\rtimes S_4
% in TeX

G:=Group("C5xC2^2:S4");
// GroupNames label

G:=SmallGroup(480,1200);
// by ID

G=gap.SmallGroup(480,1200);
# by ID

G:=PCGroup([7,-2,-5,-3,-2,2,-2,2,422,1683,185,1054,333,10085,1531,5886,608]);
// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=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,f*c*f^-1=b*c=c*b,b*d=d*b,b*e=e*b,f*b*f^-1=g*b*g=c,c*d=d*c,c*e=e*c,g*c*g=b,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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