Copied to
clipboard

G = C22.2D20  order 160 = 25·5

1st non-split extension by C22 of D20 acting via D20/D10=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C22.2D20, C23.1D10, C5⋊3(C23⋊C4), C22⋊C4⋊1D5, (C2×Dic5)⋊1C4, (C2×C10).27D4, (C22×D5)⋊1C4, C23.D5⋊1C2, C22.3(C4×D5), C22.8(C5⋊D4), C2.4(D10⋊C4), C10.13(C22⋊C4), (C22×C10).5C22, (C5×C22⋊C4)⋊1C2, (C2×C5⋊D4).1C2, (C2×C10).21(C2×C4), SmallGroup(160,13)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C2×C10 — C22.2D20
Chief series
C1 — C5 — C10 — C2×C10 — C22×C10 — C2×C5⋊D4 — C22.2D20
Lower central
C5 — C10 — C2×C10 — C22.2D20
Upper central
C1 — C2 — C23 — C22⋊C4

Generators and relations for C22.2D20
 G = < a,b,c,d | a2=b2=c20=1, d2=a, cac-1=ab=ba, ad=da, bc=cb, bd=db, dcd-1=ac-1 >

C1
C2
2C2
2C2
2C2
20C2
C5
C22
C22
C22
4C4
4C22
10C22
10C4
20C4
20C22
C10
2C10
2C10
2C10
4D5
C23
2C2×C4
5C23
5C2×C4
10C2×C4
10D4
10D4
C2×C10
C2×C10
C2×C10
2Dic5
2D10
4Dic5
4D10
4C2×C10
4C20
C22⋊C4
5C22⋊C4
5C2×D4
C22×D5
C22×C10
C2×Dic5
2C2×C20
2C5⋊D4
2C5⋊D4
2C2×Dic5
5C23⋊C4
C2×C5⋊D4
C23.D5
C5×C22⋊C4
C22.2D20

Smallest permutation representation of C22.2D20
►On 40 points
Generators in S40
(1 37)(3 39)(5 21)(7 23)(9 25)(11 27)(13 29)(15 31)(17 33)(19 35)

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

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

31 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D4E5A5B10A···10F10G10H10I10J20A···20H
order122222444445510···101010101020···20
size112222044202020222···244444···4

31 irreducible representations

dim11111122222244
type+++++++++
imageC1C2C2C2C4C4D4D5D10C4×D5D20C5⋊D4C23⋊C4C22.2D20
kernelC22.2D20C23.D5C5×C22⋊C4C2×C5⋊D4C2×Dic5C22×D5C2×C10C22⋊C4C23C22C22C22C5C1
# reps11112222244414

Matrix representation of C22.2D20 ►in GL4(𝔽41) generated by

40000
04000
0010
0001
,
40000
04000
00400
00040
,
0001
00406
351800
232000
,
351800
23600
0001
0010
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,1,0,0,0,0,1],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[0,0,35,23,0,0,18,20,0,40,0,0,1,6,0,0],[35,23,0,0,18,6,0,0,0,0,0,1,0,0,1,0] >;
 

C22.2D20 in GAP, Magma, Sage, TeX 

C_2^2._2D_{20}
 
% in TeX
 
G:=Group("C2^2.2D20");
 
// GroupNames label
 
G:=SmallGroup(160,13);
 
// by ID
 
G=gap.SmallGroup(160,13);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,297,4613]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^2=b^2=c^20=1,d^2=a,c*a*c^-1=a*b=b*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=a*c^-1>;
 
// generators/relations
 

׿
×
⋊
ℤ
𝔽
○
≀
ℚ
◁