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G = C5×C4○D20  order 400 = 24·52

Direct product of C5 and C4○D20

direct product, metabelian, supersoluble, monomial

Aliases: C5×C4○D20, D20⋊5C10, C20.68D10, Dic10⋊5C10, C102.29C22, (C2×C20)⋊7D5, (C10×C20)⋊6C2, (C2×C20)⋊4C10, (D5×C20)⋊9C2, (C4×D5)⋊4C10, C5⋊D4⋊3C10, (C5×D20)⋊11C2, C4.16(D5×C10), C52⋊8(C4○D4), C20.12(C2×C10), D10.1(C2×C10), (C2×C10).17D10, C22.2(D5×C10), (C5×Dic10)⋊11C2, C10.4(C22×C10), (C5×C10).22C23, (C5×C20).41C22, Dic5.2(C2×C10), C10.43(C22×D5), (D5×C10).14C22, (C5×Dic5).23C22, C5⋊1(C5×C4○D4), (C2×C4)⋊3(C5×D5), C2.5(D5×C2×C10), (C5×C5⋊D4)⋊7C2, (C2×C10).13(C2×C10), SmallGroup(400,184)

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 2A2B2C2D4A4B4C4D4E5A5B5C5D5E···5N10A10B10C10D10E···10AL10AM···10AT20A···20H20I···20AZ20BA···20BH
order122224444455555···51010101010···1010···1020···2020···2020···20
size1121010112101011112···211112···210···101···12···210···10

130 irreducible representations

dim1111111111112222222222
type+++++++++
imageC1C2C2C2C2C2C5C10C10C10C10C10D5C4○D4D10D10C5×D5C4○D20C5×C4○D4D5×C10D5×C10C5×C4○D20
kernelC5×C4○D20C5×Dic10D5×C20C5×D20C5×C5⋊D4C10×C20C4○D20Dic10C4×D5D20C5⋊D4C2×C20C2×C20C52C20C2×C10C2×C4C5C5C4C22C1
# reps112121448484224288816832

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

180
018
,
320
032
,
366
08
,
366
375
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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