ES+(2,2):2D5 - GroupNames

G = 2₊¹⁺⁴⋊₂D₅ order 320 = 2⁶·5

2^nd semidirect product of 2₊¹⁺⁴ and D₅ acting via D₅/C₅=C₂

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: 2₊¹⁺⁴⋊₂D₅, C₅⋊₃C₂≀C₂², (C₂×C₂₀)⋊₃D₄, (C₂²×C₁₀)⋊₄D₄, (C₂×D₄).₈₅D₁₀, C₂³⋊₂(C₅⋊D₄), C₂³⋊D₁₀⋊₁₉C₂, C₁₀.₈₃C₂²≀C₂, C₂³⋊Dic₅⋊₁₀C₂, (C₂³×D₅)⋊₂C₂², C₂³.D₅⋊₇C₂², C₂³.₆(C₂²×D₅), (C₂²×C₁₀).₆C₂³, (C₅×2₊¹⁺⁴)⋊₆C₂, (D₄×C₁₀).₁₇₉C₂², C₂.₁₇(C₂⁴⋊₂D₅), (C₂×C₄)⋊₂(C₅⋊D₄), (C₂×C₁₀).₄₅(C₂×D₄), C₂².₁₇(C₂×C₅⋊D₄), SmallGroup(320,871)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — 2₊¹⁺⁴⋊₂D₅

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂³×D₅ — C₂³⋊D₁₀ — 2₊¹⁺⁴⋊₂D₅

Lower central

C₅ — C₁₀ — C₂²×C₁₀ — 2₊¹⁺⁴⋊₂D₅

Upper central

C₁ — C₂ — C₂³ — 2₊¹⁺⁴

Generators and relations for 2₊¹⁺⁴⋊₂D₅
G = < a,b,c,d,e,f | a⁴=b²=d²=e⁵=f²=1, c²=a², bab=a^-1, ac=ca, ad=da, ae=ea, faf=a^-1cd, fcf=bc=cb, fdf=bd=db, be=eb, bf=fb, dcd=a²c, ce=ec, de=ed, fef=e^-1 >

Subgroups: 910 in 198 conjugacy classes, 43 normal (10 characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, Q₈, C₂³, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, C₂×D₄, C₂×D₄, C₄○D₄, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂²≀C₂, 2₊¹⁺⁴, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₅×Q₈, C₂²×D₅, C₂²×C₁₀, C₂²×C₁₀, C₂²×C₁₀, C₂≀C₂², D₁₀⋊C₄, C₂³.D₅, C₂×C₅⋊D₄, D₄×C₁₀, D₄×C₁₀, C₅×C₄○D₄, C₂³×D₅, C₂³⋊Dic₅, C₂³⋊D₁₀, C₅×2₊¹⁺⁴, 2₊¹⁺⁴⋊₂D₅
Quotients: C₁, C₂, C₂², D₄, C₂³, D₅, C₂×D₄, D₁₀, C₂²≀C₂, C₅⋊D₄, C₂²×D₅, C₂≀C₂², C₂×C₅⋊D₄, C₂⁴⋊₂D₅, 2₊¹⁺⁴⋊₂D₅

Smallest permutation representation of 2₊¹⁺⁴⋊₂D₅
►On 40 points

Generators in S₄₀

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

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

(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)(21 31)(22 35)(23 34)(24 33)(25 32)(26 36)(27 40)(28 39)(29 38)(30 37)

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

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

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

50 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	2H	2I	4A	4B	4C	4D	4E	4F	5A	5B	10A	10B	10C	···	10T	20A	···	20L
order	1	2	2	2	2	2	2	2	2	2	4	4	4	4	4	4	5	5	10	10	10	···	10	20	···	20
size	1	1	2	2	2	4	4	4	20	20	4	4	4	40	40	40	2	2	2	2	4	···	4	4	···	4

50 irreducible representations

dim

type

image

C₁

C₂

D₄

D₅

D₁₀

C₅⋊D₄

C₂≀C₂²

2₊¹⁺⁴⋊₂D₅

kernel

2₊¹⁺⁴⋊₂D₅

C₂³⋊Dic₅

C₂³⋊D₁₀

C₅×2₊¹⁺⁴

C₂×C₂₀

C₂²×C₁₀

2₊¹⁺⁴

C₂×D₄

C₂×C₄

C₂³

C₅

C₁

# reps

Matrix representation of 2₊¹⁺⁴⋊₂D₅ ►in GL₆(𝔽₄₁)

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

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

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

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

3	7	0	0	0	0
7	3	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

40	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	0	40
0	0	0	0	40	0

G:=sub<GL(6,GF(41))| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,40,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,40,0],[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],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1,0,0,0],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,1,0,0,0],[3,7,0,0,0,0,7,3,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,40,0,0,0,0,40,0] >;

2₊¹⁺⁴⋊₂D₅ in GAP, Magma, Sage, TeX

2_+^{1+4}\rtimes_2D_5

% in TeX

G:=Group("ES+(2,2):2D5");

// GroupNames label

G:=SmallGroup(320,871);

// by ID

G=gap.SmallGroup(320,871);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,253,254,570,438,12550]);

// Polycyclic

G:=Group<a,b,c,d,e,f|a^4=b^2=d^2=e^5=f^2=1,c^2=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,f*a*f=a^-1*c*d,f*c*f=b*c=c*b,f*d*f=b*d=d*b,b*e=e*b,b*f=f*b,d*c*d=a^2*c,c*e=e*c,d*e=e*d,f*e*f=e^-1>;

// generators/relations

G = 2+ 1+4⋊2D5 order 320 = 26·5

2nd semidirect product of 2+ 1+4 and D5 acting via D5/C5=C2

G = 2₊¹⁺⁴⋊₂D₅ order 320 = 2⁶·5

2^nd semidirect product of 2₊¹⁺⁴ and D₅ acting via D₅/C₅=C₂