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## G = C5×C22≀C2order 160 = 25·5

### Direct product of C5 and C22≀C2

direct product, metabelian, nilpotent (class 2), monomial, 2-elementary

Series: Derived Chief Lower central Upper central

 Derived series C1 — C22 — C5×C22≀C2
 Chief series C1 — C2 — C22 — C2×C10 — C22×C10 — D4×C10 — C5×C22≀C2
 Lower central C1 — C22 — C5×C22≀C2
 Upper central C1 — C2×C10 — C5×C22≀C2

Generators and relations for C5×C22≀C2
G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >

Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C4, C22, C22, C22, C5, C2×C4, D4, C23, C23, C23, C10, C10, C22⋊C4, C2×D4, C24, C20, C2×C10, C2×C10, C2×C10, C22≀C2, C2×C20, C5×D4, C22×C10, C22×C10, C22×C10, C5×C22⋊C4, D4×C10, C23×C10, C5×C22≀C2
Quotients: C1, C2, C22, C5, D4, C23, C10, C2×D4, C2×C10, C22≀C2, C5×D4, C22×C10, D4×C10, C5×C22≀C2

Smallest permutation representation of C5×C22≀C2
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 21)(2 22)(3 23)(4 24)(5 25)(6 14)(7 15)(8 11)(9 12)(10 13)(16 36)(17 37)(18 38)(19 39)(20 40)(26 34)(27 35)(28 31)(29 32)(30 33)
(1 27)(2 28)(3 29)(4 30)(5 26)(6 40)(7 36)(8 37)(9 38)(10 39)(11 17)(12 18)(13 19)(14 20)(15 16)(21 35)(22 31)(23 32)(24 33)(25 34)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 40)(7 36)(8 37)(9 38)(10 39)(11 17)(12 18)(13 19)(14 20)(15 16)(21 27)(22 28)(23 29)(24 30)(25 26)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 20)(7 16)(8 17)(9 18)(10 19)(11 37)(12 38)(13 39)(14 40)(15 36)(26 34)(27 35)(28 31)(29 32)(30 33)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 34)(7 35)(8 31)(9 32)(10 33)(11 22)(12 23)(13 24)(14 25)(15 21)(16 27)(17 28)(18 29)(19 30)(20 26)

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

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

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

C5×C22≀C2 is a maximal subgroup of
C242Dic5  C242D10  C24.56D10  C24.32D10  C243D10  C244D10  C24.33D10  C24.34D10  C24.35D10  C245D10  C24.36D10  C5×D42

70 conjugacy classes

 class 1 2A 2B 2C 2D ··· 2I 2J 4A 4B 4C 5A 5B 5C 5D 10A ··· 10L 10M ··· 10AJ 10AK 10AL 10AM 10AN 20A ··· 20L order 1 2 2 2 2 ··· 2 2 4 4 4 5 5 5 5 10 ··· 10 10 ··· 10 10 10 10 10 20 ··· 20 size 1 1 1 1 2 ··· 2 4 4 4 4 1 1 1 1 1 ··· 1 2 ··· 2 4 4 4 4 4 ··· 4

70 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 type + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 C5×D4 kernel C5×C22≀C2 C5×C22⋊C4 D4×C10 C23×C10 C22≀C2 C22⋊C4 C2×D4 C24 C2×C10 C22 # reps 1 3 3 1 4 12 12 4 6 24

Matrix representation of C5×C22≀C2 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 40 0 0 0 0 40 0 0 0 0 1
,
 40 0 0 0 0 40 0 0 0 0 1 0 0 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 1 0 0 0 0 1 0 0 0 0 40 0 0 0 0 40
,
 0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;

C5×C22≀C2 in GAP, Magma, Sage, TeX

C_5\times C_2^2\wr C_2
% in TeX

G:=Group("C5xC2^2wrC2");
// GroupNames label

G:=SmallGroup(160,181);
// by ID

G=gap.SmallGroup(160,181);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-2,505,1514]);
// Polycyclic

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

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