C2^3.2D20 - GroupNames

G = C₂³.₂D₂₀ order 320 = 2⁶·5

2^nd non-split extension by C₂³ of D₂₀ acting via D₂₀/C₅=D₄

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C₂³.₂D₂₀, (C₂×D₂₀)⋊₄C₄, C₂³⋊C₄⋊₂D₅, (C₄×Dic₅)⋊₂C₄, (C₂×D₄).₄D₁₀, C₅⋊₃(C₄²⋊C₄), C₂₀⋊D₄.₁C₂, C₂³⋊Dic₅⋊₁C₂, (D₄×C₁₀).₄C₂², (C₂²×C₁₀).₁₁D₄, C₂³.₄(C₅⋊D₄), C₁₀.₃₁(C₂³⋊C₄), C₂².₁₁(D₁₀⋊C₄), C₂.₁₁(C₂³.₁D₁₀), (C₂×C₄).₂(C₄×D₅), (C₅×C₂³⋊C₄)⋊₂C₂, (C₂×C₂₀).₂(C₂×C₄), (C₂×C₁₀).₆₈(C₂²⋊C₄), SmallGroup(320,32)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₂×C₂₀ — C₂³.₂D₂₀

Chief series

C₁ — C₅ — C₁₀ — C₂×C₁₀ — C₂²×C₁₀ — D₄×C₁₀ — C₂₀⋊D₄ — C₂³.₂D₂₀

Lower central

C₅ — C₁₀ — C₂×C₁₀ — C₂×C₂₀ — C₂³.₂D₂₀

Upper central

C₁ — C₂ — C₂² — C₂×D₄ — C₂³⋊C₄

Generators and relations for C₂³.₂D₂₀
G = < a,b,c,d,e | a²=b²=c²=d²⁰=1, e²=a, dad^-1=ab=ba, ac=ca, ae=ea, dbd^-1=ebe^-1=bc=cb, cd=dc, ce=ec, ede^-1=ad^-1 >

Subgroups: 526 in 86 conjugacy classes, 21 normal (all characteristic)
C₁, C₂, C₂, C₄, C₂², C₂², C₅, C₂×C₄, C₂×C₄, D₄, C₂³, C₂³, D₅, C₁₀, C₁₀, C₄², C₂²⋊C₄, C₂×D₄, C₂×D₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₂³⋊C₄, C₂³⋊C₄, C₄⋊₁D₄, D₂₀, C₂×Dic₅, C₅⋊D₄, C₂×C₂₀, C₂×C₂₀, C₅×D₄, C₂²×D₅, C₂²×C₁₀, C₄²⋊C₄, C₄×Dic₅, C₂³.D₅, C₅×C₂²⋊C₄, C₂×D₂₀, C₂×C₅⋊D₄, D₄×C₁₀, C₂³⋊Dic₅, C₅×C₂³⋊C₄, C₂₀⋊D₄, C₂³.₂D₂₀
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, D₄, D₅, C₂²⋊C₄, D₁₀, C₂³⋊C₄, C₄×D₅, D₂₀, C₅⋊D₄, C₄²⋊C₄, D₁₀⋊C₄, C₂³.₁D₁₀, C₂³.₂D₂₀

Smallest permutation representation of C₂³.₂D₂₀
►On 40 points

Generators in S₄₀

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

(2 17)(4 19)(6 11)(8 13)(10 15)(22 32)(24 34)(26 36)(28 38)(30 40)

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

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

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

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

35 conjugacy classes

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

35 irreducible representations

dim	1	1	1	1	1	1	2	2	2	2	2	2	4	4	4	8
type	+	+	+	+			+	+	+		+		+	+		+
image	C₁	C₂	C₂	C₂	C₄	C₄	D₄	D₅	D₁₀	C₄×D₅	D₂₀	C₅⋊D₄	C₂³⋊C₄	C₄²⋊C₄	C₂³.₁D₁₀	C₂³.₂D₂₀
kernel	C₂³.₂D₂₀	C₂³⋊Dic₅	C₅×C₂³⋊C₄	C₂₀⋊D₄	C₄×Dic₅	C₂×D₂₀	C₂²×C₁₀	C₂³⋊C₄	C₂×D₄	C₂×C₄	C₂³	C₂³	C₁₀	C₅	C₂	C₁
# reps	1	1	1	1	2	2	2	2	2	4	4	4	1	2	4	2

Matrix representation of C₂³.₂D₂₀ ►in GL₆(𝔽₄₁)

40	0	0	0	0	0
0	40	0	0	0	0
0	0	40	32	0	0
0	0	0	1	0	0
0	0	18	40	0	1
0	0	18	40	1	0

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	23	0	40	0
0	0	23	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

0	32	0	0	0	0
9	22	0	0	0	0
0	0	40	0	9	0
0	0	0	0	40	1
0	0	0	0	1	0
0	0	18	40	1	0

0	32	0	0	0	0
32	0	0	0	0	0
0	0	40	0	9	0
0	0	23	0	40	40
0	0	18	40	1	0
0	0	0	0	1	0

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,18,18,0,0,32,1,40,40,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,23,23,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40],[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],[0,9,0,0,0,0,32,22,0,0,0,0,0,0,40,0,0,18,0,0,0,0,0,40,0,0,9,40,1,1,0,0,0,1,0,0],[0,32,0,0,0,0,32,0,0,0,0,0,0,0,40,23,18,0,0,0,0,0,40,0,0,0,9,40,1,1,0,0,0,40,0,0] >;

C₂³.₂D₂₀ in GAP, Magma, Sage, TeX

C_2^3._2D_{20}

% in TeX

G:=Group("C2^3.2D20");

// GroupNames label

G:=SmallGroup(320,32);

// by ID

G=gap.SmallGroup(320,32);

# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,141,36,422,1123,794,297,136,851,12550]);

// Polycyclic

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

// generators/relations

G = C23.2D20 order 320 = 26·5

2nd non-split extension by C23 of D20 acting via D20/C5=D4

G = C₂³.₂D₂₀ order 320 = 2⁶·5

2^nd non-split extension by C₂³ of D₂₀ acting via D₂₀/C₅=D₄