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## G = C2×C5⋊D20order 400 = 24·52

### Direct product of C2 and C5⋊D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — C2×C5⋊D20
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C5⋊D20 — C2×C5⋊D20
 Lower central C52 — C5×C10 — C2×C5⋊D20
 Upper central C1 — C22

Generators and relations for C2×C5⋊D20
G = < a,b,c,d | a2=b5=c20=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Subgroups: 1004 in 140 conjugacy classes, 40 normal (18 characteristic)
C1, C2, C2, C2, C4, C22, C22, C5, C5, C2×C4, D4, C23, D5, C10, C10, C10, C2×D4, Dic5, C20, D10, D10, C2×C10, C2×C10, C52, D20, C2×Dic5, C5⋊D4, C2×C20, C22×D5, C22×D5, C22×C10, C5×D5, C5⋊D5, C5×C10, C5×C10, C2×D20, C2×C5⋊D4, C5×Dic5, D5×C10, D5×C10, C2×C5⋊D5, C2×C5⋊D5, C102, C5⋊D20, C10×Dic5, D5×C2×C10, C22×C5⋊D5, C2×C5⋊D20
Quotients: C1, C2, C22, D4, C23, D5, C2×D4, D10, D20, C5⋊D4, C22×D5, C2×D20, C2×C5⋊D4, D52, C5⋊D20, C2×D52, C2×C5⋊D20

Smallest permutation representation of C2×C5⋊D20
On 40 points
Generators in S40
(1 33)(2 34)(3 35)(4 36)(5 37)(6 38)(7 39)(8 40)(9 21)(10 22)(11 23)(12 24)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)
(1 13 5 17 9)(2 10 18 6 14)(3 15 7 19 11)(4 12 20 8 16)(21 33 25 37 29)(22 30 38 26 34)(23 35 27 39 31)(24 32 40 28 36)
(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 27)(2 26)(3 25)(4 24)(5 23)(6 22)(7 21)(8 40)(9 39)(10 38)(11 37)(12 36)(13 35)(14 34)(15 33)(16 32)(17 31)(18 30)(19 29)(20 28)

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

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

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

58 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10L 10M ··· 10X 10Y ··· 10AF 20A ··· 20H order 1 2 2 2 2 2 2 2 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 ··· 10 20 ··· 20 size 1 1 1 1 10 10 50 50 10 10 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 10 ··· 10 10 ··· 10

58 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + + + + + + + image C1 C2 C2 C2 C2 D4 D5 D5 D10 D10 D10 D20 C5⋊D4 D52 C5⋊D20 C2×D52 kernel C2×C5⋊D20 C5⋊D20 C10×Dic5 D5×C2×C10 C22×C5⋊D5 C5×C10 C2×Dic5 C22×D5 Dic5 D10 C2×C10 C10 C10 C22 C2 C2 # reps 1 4 1 1 1 2 2 2 4 4 4 8 8 4 8 4

Matrix representation of C2×C5⋊D20 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 40 0 0 0 0 1 6 0 0 0 0 0 0 1 0 0 0 0 0 0 1
,
 1 25 0 0 0 0 36 40 0 0 0 0 0 0 35 23 0 0 0 0 18 6 0 0 0 0 0 0 0 40 0 0 0 0 1 6
,
 1 0 0 0 0 0 36 40 0 0 0 0 0 0 0 40 0 0 0 0 40 0 0 0 0 0 0 0 1 6 0 0 0 0 0 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,40,6,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,25,40,0,0,0,0,0,0,35,18,0,0,0,0,23,6,0,0,0,0,0,0,0,1,0,0,0,0,40,6],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,6,40] >;

C2×C5⋊D20 in GAP, Magma, Sage, TeX

C_2\times C_5\rtimes D_{20}
% in TeX

G:=Group("C2xC5:D20");
// GroupNames label

G:=SmallGroup(400,177);
// by ID

G=gap.SmallGroup(400,177);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,55,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^5=c^20=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations

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