C2^3.3D20 - GroupNames

G = C₂³.₃D₂₀ order 320 = 2⁶·5

3^rd non-split extension by C₂³ of D₂₀ acting via D₂₀/C₅=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₃D₂₀, C₅⋊₄C₂≀C₄, (C₂×D₄).₅D₁₀, C₄.D₄⋊₅D₅, (C₂×C₂₀).₁₃D₄, (C₂³×D₅)⋊₂C₄, C₂³.D₅⋊₃C₄, C₂³.₃(C₄×D₅), C₂³⋊Dic₅⋊₇C₂, C₂³⋊D₁₀.₄C₂, (C₂²×C₁₀).₁₂D₄, C₁₀.₃₂(C₂³⋊C₄), (D₄×C₁₀).₁₇₀C₂², C₂².₁₂(D₁₀⋊C₄), C₂.₁₂(C₂³.₁D₁₀), (C₂×C₄).₁(C₅⋊D₄), (C₅×C₄.D₄)⋊₁₁C₂, (C₂²×C₁₀).₃(C₂×C₄), (C₂×C₁₀).₆₉(C₂²⋊C₄), SmallGroup(320,33)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂²×C₁₀ — C₂³.₃D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — D₄×C₁₀ — C₂³⋊D₁₀ — C₂³.₃D₂₀

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — C₂³.₃D₂₀

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₄.D₄

Generators and relations for C₂³.₃D₂₀
G = < a,b,c,d,e | a²=b²=c²=1, d²⁰=c, e²=a, ab=ba, ac=ca, dad^-1=abc, ae=ea, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=acd¹⁹ >

Subgroups: 606 in 94 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₈, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₂²⋊C₄, M₄(2), C₂×D₄, C₂×D₄, C₂⁴, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₄.D₄, C₂²≀C₂, C₄₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₂≀C₄, D₁₀⋊C₄, C₂³.D₅, C₂³.D₅, C₅×M₄(2), C₂×C₅⋊D₄, D₄×C₁₀, C₂³×D₅, C₂³⋊Dic₅, C₅×C₄.D₄, C₂³⋊D₁₀, C₂³.₃D₂₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₂≀C₄, D₁₀⋊C₄, C₂³.₁D₁₀, C₂³.₃D₂₀

Smallest permutation representation of C₂³.₃D₂₀
►On 40 points

Generators in S₄₀

(2 22)(3 23)(6 26)(7 27)(10 30)(11 31)(14 34)(15 35)(18 38)(19 39)

(2 22)(4 24)(6 26)(8 28)(10 30)(12 32)(14 34)(16 36)(18 38)(20 40)

(1 21)(2 22)(3 23)(4 24)(5 25)(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)

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

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

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

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

35 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	4A	4B	4C	4D	5A	5B	8A	8B	10A	10B	10C	10D	10E	10F	10G	10H	20A	20B	20C	20D	40A	···	40H
order	1	2	2	2	2	2	2	4	4	4	4	5	5	8	8	10	10	10	10	10	10	10	10	20	20	20	20	40	···	40
size	1	1	2	4	4	20	20	4	40	40	40	2	2	8	8	2	2	4	4	8	8	8	8	4	4	4	4	8	···	8

35 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+	+			+	+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₄	D₅	D₁₀	C₅⋊D₄	C₄×D₅	D₂₀	C₂³⋊C₄	C₂≀C₄	C₂³.₁D₁₀	C₂³.₃D₂₀
kernel	C₂³.₃D₂₀	C₂³⋊Dic₅	C₅×C₄.D₄	C₂³⋊D₁₀	C₂³.D₅	C₂³×D₅	C₂×C₂₀	C₂²×C₁₀	C₄.D₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	1	1	2	2	4	4	4	1	2	4	2

Matrix representation of C₂³.₃D₂₀ ►in GL₆(𝔽₄₁)

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	1	0	40	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	1	1	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

3	23	0	0	0	0
21	24	0	0	0	0
0	0	0	0	1	0
0	0	1	1	40	39
0	0	0	40	0	0
0	0	0	1	0	40

17	23	0	0	0	0
7	24	0	0	0	0
0	0	0	0	1	0
0	0	1	1	40	39
0	0	1	0	0	0
0	0	0	1	0	40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,1,0,0,0,40,0,0,0,0,0,0,1,40,0,0,0,0,0,40],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,1,0,1,0,0,0,0,40,0,0,0,0,0,0,40],[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],[3,21,0,0,0,0,23,24,0,0,0,0,0,0,0,1,0,0,0,0,0,1,40,1,0,0,1,40,0,0,0,0,0,39,0,40],[17,7,0,0,0,0,23,24,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0,1,0,0,1,40,0,0,0,0,0,39,0,40] >;

C₂³.₃D₂₀ in GAP, Magma, Sage, TeX

C_2^3._3D_{20}

% in TeX

G:=Group("C2^3.3D20");

// GroupNames label

G:=SmallGroup(320,33);

// by ID

G=gap.SmallGroup(320,33);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,184,346,297,851,12550]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^2=c^2=1,d^20=c,e^2=a,a*b=b*a,a*c=c*a,d*a*d^-1=a*b*c,a*e=e*a,d*b*d^-1=e*b*e^-1=b*c=c*b,c*d=d*c,c*e=e*c,e*d*e^-1=a*c*d^19>;

// generators/relations

G = C23.3D20 order 320 = 26·5

3rd non-split extension by C23 of D20 acting via D20/C5=D4

G = C₂³.₃D₂₀ order 320 = 2⁶·5

3^rd non-split extension by C₂³ of D₂₀ acting via D₂₀/C₅=D₄