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G = D5×C4○D4  order 160 = 25·5

Direct product of D5 and C4○D4

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D5×C4○D4, D4⋊7D10, Q8⋊6D10, D20⋊10C22, C20.25C23, C10.11C24, D10.6C23, Dic10⋊10C22, Dic5.17C23, (D4×D5)⋊6C2, (C2×C4)⋊7D10, (Q8×D5)⋊6C2, C4○D20⋊7C2, D4⋊2D5⋊6C2, (C2×C20)⋊4C22, Q8⋊2D5⋊6C2, (C5×D4)⋊8C22, (C4×D5)⋊7C22, C5⋊D4⋊4C22, (C5×Q8)⋊7C22, (C2×C10).3C23, C4.25(C22×D5), C2.12(C23×D5), C22.2(C22×D5), (C2×Dic5)⋊10C22, (C22×D5).34C22, (C2×C4×D5)⋊6C2, C5⋊4(C2×C4○D4), (C5×C4○D4)⋊3C2, SmallGroup(160,223)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D5×C4○D4
Chief series
C1 — C5 — C10 — D10 — C22×D5 — C2×C4×D5 — D5×C4○D4
Lower central
C5 — C10 — D5×C4○D4
Upper central
C1 — C4 — C4○D4

Generators and relations for D5×C4○D4
 G = < a,b,c,d,e | a5=b2=c4=e2=1, d2=c2, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, cd=dc, ce=ec, ede=c2d >

Subgroups: 472 in 164 conjugacy classes, 87 normal (16 characteristic)
C1, C2, C2, C4, C4, C4, C22, C22, C5, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, D5, D5, C10, C10, C22×C4, C2×D4, C2×Q8, C4○D4, C4○D4, Dic5, Dic5, C20, C20, D10, D10, D10, C2×C10, C2×C4○D4, Dic10, C4×D5, C4×D5, D20, C2×Dic5, C5⋊D4, C2×C20, C5×D4, C5×Q8, C22×D5, C2×C4×D5, C4○D20, D4×D5, D4⋊2D5, Q8×D5, Q8⋊2D5, C5×C4○D4, D5×C4○D4
Quotients: C1, C2, C22, C23, D5, C4○D4, C24, D10, C2×C4○D4, C22×D5, C23×D5, D5×C4○D4

Smallest permutation representation of D5×C4○D4
►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 8)(2 7)(3 6)(4 10)(5 9)(11 16)(12 20)(13 19)(14 18)(15 17)(21 26)(22 30)(23 29)(24 28)(25 27)(31 36)(32 40)(33 39)(34 38)(35 37)

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

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

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

D5×C4○D4 is a maximal subgroup of
 C42⋊D10  D5⋊C4≀C2  C4○D4⋊F5  C4○D20⋊C4  D4⋊F5⋊C2  C20.72C24  Q16⋊D10  SD16⋊D10  D40⋊C22  Dic5.21C24  Dic5.22C24  D5.2+ 1+4  C10.C25  D20.37C23  D20.39C23  D20⋊24D6  D30.C23  D20⋊16D6
D5×C4○D4 is a maximal quotient of
 C42.88D10  C42.188D10  C42⋊8D10  C42⋊10D10  C42.93D10  C42.94D10  C42.95D10  C42.96D10  C42.97D10  C42.98D10  C4×D4⋊2D5  C42.102D10  D4⋊5Dic10  C42.104D10  C4×D4×D5  C42⋊12D10  C42.228D10  D4⋊5D20  C42⋊16D10  C42.229D10  C42.113D10  C42.114D10  C42.122D10  Q8⋊5Dic10  C4×Q8×D5  C4×Q8⋊2D5  Q8⋊5D20  C42.232D10  C42.131D10  C42.132D10  C20⋊(C4○D4)  Dic10⋊20D4  C4⋊C4.178D10  C10.342+ 1+4  C4⋊C4⋊21D10  C10.402+ 1+4  D20⋊20D4  C10.422+ 1+4  C10.432+ 1+4  C10.442+ 1+4  C10.452+ 1+4  (Q8×Dic5)⋊C2  C22⋊Q8⋊25D5  C4⋊C4⋊26D10  D20⋊22D4  Dic10⋊22D4  C10.522+ 1+4  C10.532+ 1+4  C10.202- 1+4  C10.212- 1+4  C10.222- 1+4  C10.232- 1+4  C4⋊C4.197D10  C10.802- 1+4  C10.1212+ 1+4  C10.822- 1+4  C4⋊C4⋊28D10  C10.612+ 1+4  C10.1222+ 1+4  C10.622+ 1+4  C10.632+ 1+4  C10.642+ 1+4  C10.842- 1+4  C10.662+ 1+4  C10.672+ 1+4  C42.233D10  C42.137D10  C42.138D10  C42.139D10  D20⋊10D4  Dic10⋊10D4  C42⋊20D10  C42⋊21D10  C42.234D10  C42.143D10  Dic10⋊7Q8  C42.236D10  D20⋊7Q8  C42.237D10  C42.150D10  C42.151D10  C42.152D10  C42.153D10  C42.154D10  C42.159D10  C42.160D10  C42⋊23D10  C42⋊24D10  C42.189D10  C42.161D10  C42.162D10  C42.163D10  C42.164D10  C10.1042- 1+4  (C2×C20)⋊15D4  C10.1452+ 1+4  C10.1072- 1+4  (C2×C20)⋊17D4  C10.1482+ 1+4  D20⋊24D6  D30.C23  D20⋊16D6

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I4A4B4C4D4E4F4G4H4I4J5A5B10A10B10C···10H20A20B20C20D20E···20J
order1222222222444444444455101010···102020202020···20
size1122255101010112225510101022224···422224···4

40 irreducible representations

dim11111111222224
type++++++++++++
imageC1C2C2C2C2C2C2C2D5C4○D4D10D10D10D5×C4○D4
kernelD5×C4○D4C2×C4×D5C4○D20D4×D5D4⋊2D5Q8×D5Q8⋊2D5C5×C4○D4C4○D4D5C2×C4D4Q8C1
# reps13333111246624

Matrix representation of D5×C4○D4 ►in GL4(𝔽41) generated by

0100
403400
0010
0001
,
0100
1000
00400
00040
,
1000
0100
00320
00032
,
40000
04000
0001
00400
,
40000
04000
0001
0010
G:=sub<GL(4,GF(41))| [0,40,0,0,1,34,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,32,0,0,0,0,32],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,1,0],[40,0,0,0,0,40,0,0,0,0,0,1,0,0,1,0] >;
 

D5×C4○D4 in GAP, Magma, Sage, TeX 

D_5\times C_4\circ D_4
 
% in TeX
 
G:=Group("D5xC4oD4");
 
// GroupNames label
 
G:=SmallGroup(160,223);
 
// by ID
 
G=gap.SmallGroup(160,223);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,86,297,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^2=c^4=e^2=1,d^2=c^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=c^2*d>;
 
// generators/relations
 

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