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G = D5xC22:C4  order 160 = 25·5

Direct product of D5 and C22:C4

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

Aliases: D5xC22:C4, D10.21D4, C23.13D10, (C2xC4):5D10, C2.1(D4xD5), D10:6(C2xC4), C22:3(C4xD5), (C2xC20):6C22, C10.17(C2xD4), (C22xD5):3C4, C23.D5:3C2, D10:C4:9C2, (C23xD5).1C2, (C2xC10).21C23, C10.19(C22xC4), (C2xDic5):5C22, C22.13(C22xD5), (C22xC10).10C22, (C22xD5).42C22, (C2xC4xD5):8C2, C2.8(C2xC4xD5), C5:2(C2xC22:C4), (C2xC10):4(C2xC4), (C5xC22:C4):8C2, SmallGroup(160,101)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — D5xC22:C4
Chief series
C1 — C5 — C10 — C2xC10 — C22xD5 — C23xD5 — D5xC22:C4
Lower central
C5 — C10 — D5xC22:C4
Upper central
C1 — C22 — C22:C4

Generators and relations for D5xC22:C4
 G = < a,b,c,d,e | a5=b2=c2=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, ece-1=cd=dc, de=ed >

Subgroups: 472 in 132 conjugacy classes, 49 normal (15 characteristic)
C1, C2, C2, C2, C4, C22, C22, C22, C5, C2xC4, C2xC4, C23, C23, D5, D5, C10, C10, C10, C22:C4, C22:C4, C22xC4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, C2xC22:C4, C4xD5, C2xDic5, C2xC20, C22xD5, C22xD5, C22xD5, C22xC10, D10:C4, C23.D5, C5xC22:C4, C2xC4xD5, C23xD5, D5xC22:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, D5, C22:C4, C22xC4, C2xD4, D10, C2xC22:C4, C4xD5, C22xD5, C2xC4xD5, D4xD5, D5xC22:C4

Smallest permutation representation of D5xC22:C4
►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 14)(2 15)(3 11)(4 12)(5 13)(6 16)(7 17)(8 18)(9 19)(10 20)(21 36)(22 37)(23 38)(24 39)(25 40)(26 31)(27 32)(28 33)(29 34)(30 35)

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

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

D5xC22:C4 is a maximal subgroup of
 (C22xF5):C4  D10:(C4:C4)  C10.(C4xD4)  C24.24D10  C24.27D10  C42:7D10  C42:10D10  C4xD4xD5  C42:11D10  D20:23D4  C42:16D10  C24:3D10  C24.33D10  C10.402+ 1+4  D20:20D4  C10.422+ 1+4  D20:21D4  C10.512+ 1+4  C10.532+ 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.612+ 1+4  C10.1222+ 1+4  C10.622+ 1+4  D20:10D4  C42:20D10  C42:21D10  C42:23D10  C42:24D10  D30.27D4  D30.45D4
D5xC22:C4 is a maximal quotient of
 (C2xC20):Q8  C22.58(D4xD5)  (C2xC4):9D20  D10:2C42  D10:2(C4:C4)  D10:7M4(2)  C22:C8:D5  C23:C4:5D5  M4(2).19D10  M4(2).21D10  (D4xD5):C4  D4:(C4xD5)  D4:2D5:C4  (Q8xD5):C4  Q8:(C4xD5)  Q8:2D5:C4  C42:D10  C24.44D10  C24.48D10  C24.12D10  C24.13D10  D30.27D4  D30.45D4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K4A4B4C4D4E4F4G4H5A5B10A···10F10G10H10I10J20A···20H
order122222222222444444445510···101010101020···20
size11112255551010222210101010222···244444···4

40 irreducible representations

dim1111111222224
type+++++++++++
imageC1C2C2C2C2C2C4D4D5D10D10C4xD5D4xD5
kernelD5xC22:C4D10:C4C23.D5C5xC22:C4C2xC4xD5C23xD5C22xD5D10C22:C4C2xC4C23C22C2
# reps1211218424284

Matrix representation of D5xC22:C4 ►in GL4(F41) generated by

0100
40600
0010
0001
,
0100
1000
00400
00040
,
40000
04000
0010
003240
,
1000
0100
00400
00040
,
9000
0900
003239
00409
G:=sub<GL(4,GF(41))| [0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,1,0,0,1,0,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,1,32,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,40,0,0,0,0,40],[9,0,0,0,0,9,0,0,0,0,32,40,0,0,39,9] >;
 

D5xC22:C4 in GAP, Magma, Sage, TeX 

D_5\times C_2^2\rtimes C_4
 
% in TeX
 
G:=Group("D5xC2^2:C4");
 
// GroupNames label
 
G:=SmallGroup(160,101);
 
// by ID
 
G=gap.SmallGroup(160,101);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,188,50,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^5=b^2=c^2=d^2=e^4=1,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,e*c*e^-1=c*d=d*c,d*e=e*d>;
 
// generators/relations
 

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