D5xC2^2wrC2 - GroupNames

G = D₅×C₂²≀C₂ order 320 = 2⁶·5

Direct product of D₅ and C₂²≀C₂

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

Aliases: D₅×C₂²≀C₂, C₂⁴⋊₁₀D₁₀, C₂²⋊₅(D₄×D₅), (C₂×D₄)⋊₁₇D₁₀, D₁₀⋊₁₁(C₂×D₄), (C₂×C₂₀)⋊₁C₂³, (D₅×C₂⁴)⋊₂C₂, C₂²⋊D₂₀⋊₈C₂, C₂³⋊D₁₀⋊₂C₂, C₂²⋊C₄⋊₂₃D₁₀, (C₂²×D₅)⋊₁₄D₄, (D₄×C₁₀)⋊₅C₂², C₂⁴⋊₂D₅⋊₅C₂, C₂³⋊₁(C₂²×D₅), (C₂×D₂₀)⋊₁₇C₂², (C₂²×C₁₀)⋊₁C₂³, (C₂³×C₁₀)⋊₈C₂², (C₂×Dic₅)⋊₂C₂³, C₁₀.₅₄(C₂²×D₄), (C₂²×D₅)⋊₂C₂³, D₁₀⋊C₄⋊₉C₂², (C₂×C₁₀).₁₃₂C₂⁴, (C₂³×D₅)⋊₂₀C₂², C₂³.D₅⋊₁₃C₂², C₂².₁₅₃(C₂³×D₅), (C₂×D₄×D₅)⋊₅C₂, C₂.₂₇(C₂×D₄×D₅), C₅⋊₂(C₂×C₂²≀C₂), (C₂×C₁₀)⋊₅(C₂×D₄), (C₂×C₄×D₅)⋊₅C₂², (D₅×C₂²⋊C₄)⋊₁C₂, (C₂×C₄)⋊₁(C₂²×D₅), (C₅×C₂²≀C₂)⋊₃C₂, (C₂×C₅⋊D₄)⋊₇C₂², (C₅×C₂²⋊C₄)⋊₃C₂², SmallGroup(320,1260)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₁₀ — D₅×C₂²≀C₂

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×D₅ — C₂³×D₅ — D₅×C₂⁴ — D₅×C₂²≀C₂

Lower central

C₅ — C₂×C₁₀ — D₅×C₂²≀C₂

Upper central

C₁ — C₂² — C₂²≀C₂

Generators and relations for D₅×C₂²≀C₂
G = < a,b,c,d,e,f,g | a⁵=b²=c²=d²=e²=f²=g²=1, bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, gcg=ce=ec, cf=fc, de=ed, gdg=df=fd, ef=fe, eg=ge, fg=gf >

Subgroups: 2990 in 662 conjugacy classes, 131 normal (16 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, C₂³, D₅, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂²⋊C₄, C₂²×C₄, C₂×D₄, C₂×D₄, C₂⁴, C₂⁴, Dic₅, C₂₀, D₁₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₁₀, C₂×C₂²⋊C₄, C₂²≀C₂, C₂²≀C₂, C₂²×D₄, C₂⁵, C₄×D₅, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×D₅, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂×C₂²≀C₂, D₁₀⋊C₄, C₂³.D₅, C₅×C₂²⋊C₄, C₂×C₄×D₅, C₂×D₂₀, D₄×D₅, C₂×C₅⋊D₄, D₄×C₁₀, C₂³×D₅, C₂³×D₅, C₂³×D₅, C₂³×C₁₀, D₅×C₂²⋊C₄, C₂²⋊D₂₀, C₂³⋊D₁₀, C₂⁴⋊₂D₅, C₅×C₂²≀C₂, C₂×D₄×D₅, D₅×C₂⁴, D₅×C₂²≀C₂
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, C₂⁴, D₁₀, C₂²≀C₂, C₂²×D₄, C₂²×D₅, C₂×C₂²≀C₂, D₄×D₅, C₂³×D₅, C₂×D₄×D₅, D₅×C₂²≀C₂

Smallest permutation representation of D₅×C₂²≀C₂
►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 5)(2 4)(7 10)(8 9)(12 15)(13 14)(17 20)(18 19)(22 25)(23 24)(27 30)(28 29)(32 35)(33 34)(37 40)(38 39)

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

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

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

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

56 conjugacy classes

class	1	2A	2B	2C	2D	···	2I	2J	2K	2L	2M	2N	2O	···	2T	2U	4A	4B	4C	4D	4E	4F	5A	5B	10A	···	10F	10G	···	10R	10S	10T	20A	···	20F
order	1	2	2	2	2	···	2	2	2	2	2	2	2	···	2	2	4	4	4	4	4	4	5	5	10	···	10	10	···	10	10	10	20	···	20
size	1	1	1	1	2	···	2	4	5	5	5	5	10	···	10	20	4	4	4	20	20	20	2	2	2	···	2	4	···	4	8	8	8	···	8

56 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

D₄×D₅

kernel

D₅×C₂²≀C₂

D₅×C₂²⋊C₄

C₂²⋊D₂₀

C₂³⋊D₁₀

C₂⁴⋊₂D₅

C₅×C₂²≀C₂

C₂×D₄×D₅

D₅×C₂⁴

C₂²×D₅

C₂²≀C₂

C₂²⋊C₄

C₂×D₄

C₂⁴

C₂²

# reps

Matrix representation of D₅×C₂²≀C₂ ►in GL₆(𝔽₄₁)

6	1	0	0	0	0
40	0	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	6	0	0	0	0
0	40	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	1	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	1

1	0	0	0	0	0
0	1	0	0	0	0
0	0	1	0	0	0
0	0	0	1	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	40	0
0	0	0	0	0	40

1	0	0	0	0	0
0	1	0	0	0	0
0	0	0	1	0	0
0	0	1	0	0	0
0	0	0	0	0	1
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [6,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,6,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;

D₅×C₂²≀C₂ in GAP, Magma, Sage, TeX

D_5\times C_2^2\wr C_2

% in TeX

G:=Group("D5xC2^2wrC2");

// GroupNames label

G:=SmallGroup(320,1260);

// by ID

G=gap.SmallGroup(320,1260);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,570,185,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f,g|a^5=b^2=c^2=d^2=e^2=f^2=g^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,g*c*g=c*e=e*c,c*f=f*c,d*e=e*d,g*d*g=d*f=f*d,e*f=f*e,e*g=g*e,f*g=g*f>;

// generators/relations

G = D5×C22≀C2 order 320 = 26·5

Direct product of D5 and C22≀C2

G = D₅×C₂²≀C₂ order 320 = 2⁶·5

Direct product of D₅ and C₂²≀C₂