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## G = C5×C4○D20order 400 = 24·52

### Direct product of C5 and C4○D20

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C4○D20
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — D5×C20 — C5×C4○D20
 Lower central C5 — C10 — C5×C4○D20
 Upper central C1 — C20 — C2×C20

Generators and relations for C5×C4○D20
G = < a,b,c,d | a5=b4=d2=1, c10=b2, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=b2c9 >

Subgroups: 244 in 96 conjugacy classes, 46 normal (30 characteristic)
C1, C2, C2, C4, C4, C22, C22, C5, C5, C2×C4, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, C20, C20, D10, C2×C10, C2×C10, C52, Dic10, C4×D5, D20, C5⋊D4, C2×C20, C2×C20, C5×D4, C5×Q8, C5×D5, C5×C10, C5×C10, C4○D20, C5×C4○D4, C5×Dic5, C5×C20, D5×C10, C102, C5×Dic10, D5×C20, C5×D20, C5×C5⋊D4, C10×C20, C5×C4○D20
Quotients: C1, C2, C22, C5, C23, D5, C10, C4○D4, D10, C2×C10, C22×D5, C22×C10, C5×D5, C4○D20, C5×C4○D4, D5×C10, D5×C2×C10, C5×C4○D20

Smallest permutation representation of C5×C4○D20
On 40 points
Generators in S40
(1 9 17 5 13)(2 10 18 6 14)(3 11 19 7 15)(4 12 20 8 16)(21 33 25 37 29)(22 34 26 38 30)(23 35 27 39 31)(24 36 28 40 32)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(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 29)(3 28)(4 27)(5 26)(6 25)(7 24)(8 23)(9 22)(10 21)(11 40)(12 39)(13 38)(14 37)(15 36)(16 35)(17 34)(18 33)(19 32)(20 31)

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

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

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

130 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10AL 10AM ··· 10AT 20A ··· 20H 20I ··· 20AZ 20BA ··· 20BH order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 20 ··· 20 size 1 1 2 10 10 1 1 2 10 10 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 10 ··· 10 1 ··· 1 2 ··· 2 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 type + + + + + + + + + image C1 C2 C2 C2 C2 C2 C5 C10 C10 C10 C10 C10 D5 C4○D4 D10 D10 C5×D5 C4○D20 C5×C4○D4 D5×C10 D5×C10 C5×C4○D20 kernel C5×C4○D20 C5×Dic10 D5×C20 C5×D20 C5×C5⋊D4 C10×C20 C4○D20 Dic10 C4×D5 D20 C5⋊D4 C2×C20 C2×C20 C52 C20 C2×C10 C2×C4 C5 C5 C4 C22 C1 # reps 1 1 2 1 2 1 4 4 8 4 8 4 2 2 4 2 8 8 8 16 8 32

Matrix representation of C5×C4○D20 in GL2(𝔽41) generated by

 18 0 0 18
,
 32 0 0 32
,
 36 6 0 8
,
 36 6 37 5
G:=sub<GL(2,GF(41))| [18,0,0,18],[32,0,0,32],[36,0,6,8],[36,37,6,5] >;

C5×C4○D20 in GAP, Magma, Sage, TeX

C_5\times C_4\circ D_{20}
% in TeX

G:=Group("C5xC4oD20");
// GroupNames label

G:=SmallGroup(400,184);
// by ID

G=gap.SmallGroup(400,184);
# by ID

G:=PCGroup([6,-2,-2,-2,-5,-2,-5,247,794,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=d^2=1,c^10=b^2,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=b^2*c^9>;
// generators/relations

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