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G = C2xD4xD5  order 160 = 25·5

Direct product of C2, D4 and D5

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

Aliases: C2xD4xD5, C20:C23, C23:3D10, D20:7C22, D10:2C23, C10.5C24, Dic5:1C23, (C2xC4):6D10, C10:2(C2xD4), (C2xC10):C23, C5:2(C22xD4), (D4xC10):5C2, C4:1(C22xD5), (C2xD20):11C2, (C2xC20):2C22, (C23xD5):4C2, (C4xD5):3C22, (C5xD4):5C22, C5:D4:1C22, C2.6(C23xD5), C22:1(C22xD5), (C22xC10):4C22, (C2xDic5):8C22, (C22xD5):6C22, (C2xC4xD5):3C2, (C2xC5:D4):9C2, SmallGroup(160,217)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C10 — C2xD4xD5
Chief series
C1 — C5 — C10 — D10 — C22xD5 — C23xD5 — C2xD4xD5
Lower central
C5 — C10 — C2xD4xD5
Upper central
C1 — C22 — C2xD4

Generators and relations for C2xD4xD5
 G = < a,b,c,d,e | a2=b4=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 808 in 236 conjugacy classes, 97 normal (15 characteristic)
C1, C2, C2, C2, C4, C4, C22, C22, C22, C5, C2xC4, C2xC4, D4, D4, C23, C23, D5, D5, C10, C10, C10, C22xC4, C2xD4, C2xD4, C24, Dic5, C20, D10, D10, C2xC10, C2xC10, C2xC10, C22xD4, C4xD5, D20, C2xDic5, C5:D4, C2xC20, C5xD4, C22xD5, C22xD5, C22xD5, C22xC10, C2xC4xD5, C2xD20, D4xD5, C2xC5:D4, D4xC10, C23xD5, C2xD4xD5
Quotients: C1, C2, C22, D4, C23, D5, C2xD4, C24, D10, C22xD4, C22xD5, D4xD5, C23xD5, C2xD4xD5

Smallest permutation representation of C2xD4xD5
►On 40 points
Generators in S40
(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)

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

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

C2xD4xD5 is a maximal subgroup of
 (D4xD5):C4  D4:D20  D20.8D4  D20:D4  D10:6SD16  (D4xC10):C4  (C2xD4):7F5  D5:(C4.D4)  (C2xF5):D4  C2.(D4xF5)  C42:11D10  D4:5D20  C24:3D10  C24:4D10  C10.372+ 1+4  C10.382+ 1+4  D20:19D4  C10.402+ 1+4  D20:20D4  C10.1202+ 1+4  C10.1212+ 1+4  C42:18D10  D20:10D4  C42:26D10  D20:11D4  C10.1452+ 1+4  D10.C24
C2xD4xD5 is a maximal quotient of
 C24.27D10  C10.2- 1+4  C42:12D10  C42.228D10  D20:23D4  D20:24D4  Dic10:23D4  Dic10:24D4  C24.56D10  C24:3D10  C24:4D10  C24.33D10  C24.34D10  C20:(C4oD4)  C10.682- 1+4  Dic10:19D4  Dic10:20D4  C10.372+ 1+4  C4:C4:21D10  C10.382+ 1+4  C10.392+ 1+4  D20:19D4  C10.402+ 1+4  C10.732- 1+4  D20:20D4  C4:C4:26D10  C10.162- 1+4  C10.172- 1+4  D20:21D4  D20:22D4  Dic10:21D4  Dic10:22D4  C10.792- 1+4  C10.1202+ 1+4  C10.1212+ 1+4  C10.822- 1+4  C4:C4:28D10  C42.233D10  C42:18D10  C42.141D10  D20:10D4  Dic10:10D4  C42:26D10  C42.238D10  D20:11D4  Dic10:11D4  C42.171D10  C42.240D10  D20:12D4  D20:8Q8  D8:13D10  D20.29D4  D20.30D4  Q16:D10  D8:15D10  D8:11D10  D20.47D4  SD16:D10  D8:5D10  D8:6D10  D40:C22  C40.C23  D20.44D4

40 conjugacy classes

class 1 2A2B2C2D2E2F2G2H2I2J2K2L2M2N2O4A4B4C4D5A5B10A···10F10G···10N20A20B20C20D
order122222222222222244445510···1010···1020202020
size11112222555510101010221010222···24···44444

40 irreducible representations

dim1111111222224
type+++++++++++++
imageC1C2C2C2C2C2C2D4D5D10D10D10D4xD5
kernelC2xD4xD5C2xC4xD5C2xD20D4xD5C2xC5:D4D4xC10C23xD5D10C2xD4C2xC4D4C23C2
# reps1118212422844

Matrix representation of C2xD4xD5 ►in GL4(F41) generated by

40000
04000
00400
00040
,
1000
0100
00040
0010
,
1000
0100
0010
00040
,
0100
40600
0010
0001
,
04000
40000
0010
0001
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,40],[0,40,0,0,1,6,0,0,0,0,1,0,0,0,0,1],[0,40,0,0,40,0,0,0,0,0,1,0,0,0,0,1] >;
 

C2xD4xD5 in GAP, Magma, Sage, TeX 

C_2\times D_4\times D_5
 
% in TeX
 
G:=Group("C2xD4xD5");
 
// GroupNames label
 
G:=SmallGroup(160,217);
 
// by ID
 
G=gap.SmallGroup(160,217);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,159,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e|a^2=b^4=c^2=d^5=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
 
// generators/relations
 

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