D5xES+(2,2)

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

Aliases: D₅×2₊⁽¹⁺⁴⁾, D₂₀⋊₁₃C₂³, C₁₀.₁₅C₂⁵, C₂₀.₅₀C₂⁴, D₁₀.₂₅C₂⁴, Dic₁₀⋊₁₂C₂³, Dic₅.₁₀C₂⁴, C₄○D₄⋊₁₀D₁₀, (C₂×D₄)⋊₃₁D₁₀, (C₂×C₂₀)⋊₂C₂³, (C₄×D₅)⋊₃C₂³, C₅⋊D₄⋊₆C₂³, D₄⋊₆D₁₀⋊₉C₂, Q₈⋊₉(C₂²×D₅), (D₄×D₅)⋊₁₇C₂², D₄⋊₁₀(C₂²×D₅), (C₅×D₄)⋊₁₁C₂³, (C₂×C₁₀).₆C₂⁴, D₄⋊₈D₁₀⋊₁₁C₂, (Q₈×D₅)⋊₂₀C₂², (C₅×Q₈)⋊₁₀C₂³, C₂.₁₆(D₅×C₂⁴), C₄.₄₇(C₂³×D₅), C₂³⋊₂(C₂²×D₅), C₅⋊₃(C₂×2₊⁽¹⁺⁴⁾), C₄○D₂₀⋊₁₂C₂², (C₂×D₂₀)⋊₄₀C₂², (D₄×C₁₀)⋊₂₅C₂², (C₂²×C₁₀)⋊₂C₂³, (C₂×Dic₅)⋊₆C₂³, (C₂²×D₅)⋊₆C₂³, D₄⋊₂D₅⋊₁₅C₂², C₂².₃(C₂³×D₅), Q₈⋊₂D₅⋊₁₅C₂², (C₂³×D₅)⋊₁₉C₂², (C₅×2₊⁽¹⁺⁴⁾)⋊₄C₂, (C₂×D₄×D₅)⋊₂₉C₂, (D₅×C₄○D₄)⋊₇C₂, (C₂×C₄×D₅)⋊₃₆C₂², (C₂×C₄)⋊₂(C₂²×D₅), (C₅×C₄○D₄)⋊₁₀C₂², (C₂×C₅⋊D₄)⋊₃₂C₂², SmallGroup(320,1622)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₁₀ — D₅×2₊⁽¹⁺⁴⁾

Chief series

C₁ — C₅ — C₁₀ — D₁₀ — C₂²×D₅ — C₂³×D₅ — C₂×D₄×D₅ — D₅×2₊⁽¹⁺⁴⁾

Lower central

C₅ — C₁₀ — D₅×2₊⁽¹⁺⁴⁾

Upper central

C₁ — C₂ — 2₊⁽¹⁺⁴⁾

Subgroups: 3118 in 898 conjugacy classes, 445 normal (8 characteristic)
C₁, C₂, C₂^[×20], C₄^[×6], C₄^[×6], C₂²^[×9], C₂²^[×52], C₅, C₂×C₄^[×9], C₂×C₄^[×33], D₄^[×18], D₄^[×54], Q₈^[×2], Q₈^[×6], C₂³^[×6], C₂³^[×39], D₅^[×2], D₅^[×9], C₁₀, C₁₀^[×9], C₂²×C₄^[×9], C₂×D₄^[×9], C₂×D₄^[×81], C₂×Q₈^[×2], C₄○D₄^[×6], C₄○D₄^[×42], C₂⁴^[×6], Dic₅^[×6], C₂₀^[×6], D₁₀, D₁₀^[×9], D₁₀^[×36], C₂×C₁₀^[×9], C₂×C₁₀^[×6], C₂²×D₄^[×9], C₂×C₄○D₄^[×6], 2₊⁽¹⁺⁴⁾, 2₊⁽¹⁺⁴⁾^[×15], Dic₁₀^[×6], C₄×D₅^[×24], D₂₀^[×18], C₂×Dic₅^[×9], C₅⋊D₄^[×36], C₂×C₂₀^[×9], C₅×D₄^[×18], C₅×Q₈^[×2], C₂²×D₅^[×27], C₂²×D₅^[×12], C₂²×C₁₀^[×6], C₂×2₊⁽¹⁺⁴⁾, C₂×C₄×D₅^[×9], C₂×D₂₀^[×9], C₄○D₂₀^[×18], D₄×D₅^[×54], D₄⋊₂D₅^[×18], Q₈×D₅^[×2], Q₈⋊₂D₅^[×6], C₂×C₅⋊D₄^[×18], D₄×C₁₀^[×9], C₅×C₄○D₄^[×6], C₂³×D₅^[×6], C₂×D₄×D₅^[×9], D₄⋊₆D₁₀^[×9], D₅×C₄○D₄^[×6], D₄⋊₈D₁₀^[×6], C₅×2₊⁽¹⁺⁴⁾, D₅×2₊⁽¹⁺⁴⁾

Quotients:
C₁, C₂^[×31], C₂²^[×155], C₂³^[×155], D₅, C₂⁴^[×31], D₁₀^[×15], 2₊⁽¹⁺⁴⁾^[×2], C₂⁵, C₂²×D₅^[×35], C₂×2₊⁽¹⁺⁴⁾, C₂³×D₅^[×15], D₅×C₂⁴, D₅×2₊⁽¹⁺⁴⁾

Generators and relations
G = < a,b,c,d,e,f | a⁵=b²=c⁴=d²=f²=1, e²=c², bab=a^-1, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, dcd=c^-1, ce=ec, cf=fc, de=ed, df=fd, fef=c²e >

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

(11 16)(12 17)(13 18)(14 19)(15 20)(31 36)(32 37)(33 38)(34 39)(35 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)

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

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

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

Matrix representation ►G ⊆ GL₆(𝔽₄₁)

40	1	0	0	0	0
5	35	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
36	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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	40	0	0
0	0	1	0	0	0
0	0	1	1	1	2
0	0	40	0	40	40

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

40	0	0	0	0	0
0	40	0	0	0	0
0	0	0	0	40	0
0	0	1	1	1	2
0	0	1	0	0	0
0	0	40	40	0	40

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	40	40	0	40

G:=sub<GL(6,GF(41))| [40,5,0,0,0,0,1,35,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,36,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,40,0,1,0,0,0,0,0,1,40,0,0,0,0,2,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,40,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[40,0,0,0,0,0,0,40,0,0,0,0,0,0,0,1,1,40,0,0,0,1,0,40,0,0,40,1,0,0,0,0,0,2,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,40,0,0,0,1,0,40,0,0,0,0,40,0,0,0,0,0,0,40] >;

68 conjugacy classes

class	1	2A	2B	···	2J	2K	2L	2M	···	2U	4A	···	4F	4G	···	4L	5A	5B	10A	10B	10C	···	10T	20A	···	20L
order	1	2	2	···	2	2	2	2	···	2	4	···	4	4	···	4	5	5	10	10	10	···	10	20	···	20
size	1	1	2	···	2	5	5	10	···	10	2	···	2	10	···	10	2	2	2	2	4	···	4	4	···	4

68 irreducible representations

dim	1	1	1	1	1	1	2	2	2	4	8
type	+	+	+	+	+	+	+	+	+	+	+
image	C₁	C₂	C₂	C₂	C₂	C₂	D₅	D₁₀	D₁₀	2₊⁽¹⁺⁴⁾	D₅×2₊⁽¹⁺⁴⁾
kernel	D₅×2₊⁽¹⁺⁴⁾	C₂×D₄×D₅	D₄⋊₆D₁₀	D₅×C₄○D₄	D₄⋊₈D₁₀	C₅×2₊⁽¹⁺⁴⁾	2₊⁽¹⁺⁴⁾	C₂×D₄	C₄○D₄	D₅	C₁
# reps	1	9	9	6	6	1	2	18	12	2	2

In GAP, Magma, Sage, TeX

D_5\times 2_+^{(1+4)}

% in TeX

G:=Group("D5xES+(2,2)");

// GroupNames label

G:=SmallGroup(320,1622);

// by ID

G=gap.SmallGroup(320,1622);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,297,851,12550]);

// Polycyclic

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

// generators/relations

G = D₅×2₊⁽¹⁺⁴⁾ order 320 = 2⁶·5

Direct product of D₅ and 2₊⁽¹⁺⁴⁾

G = D5×2+ (1+4) order 320 = 26·5

Direct product of D5 and 2+ (1+4)

G = D₅×2₊⁽¹⁺⁴⁾ order 320 = 2⁶·5

Direct product of D₅ and 2₊⁽¹⁺⁴⁾