Copied to
clipboard

G = C20.46D4order 160 = 25·5

3rd non-split extension by C20 of D4 acting via D4/C22=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C20.46D4, C4.11D20, M4(2)⋊3D5, (C2×C4).1D10, (C2×D20).7C2, C52(C4.D4), C4.Dic52C2, C22.4(C4×D5), C4.21(C5⋊D4), (C5×M4(2))⋊7C2, (C22×D5).1C4, (C2×C20).13C22, C2.9(D10⋊C4), C10.19(C22⋊C4), (C2×C10).22(C2×C4), SmallGroup(160,30)

Series: Derived Chief Lower central Upper central

C1C2×C10 — C20.46D4
C1C5C10C20C2×C20C2×D20 — C20.46D4
C5C10C2×C10 — C20.46D4
C1C2C2×C4M4(2)

Generators and relations for C20.46D4
 G = < a,b,c | a20=c2=1, b4=a10, bab-1=cac=a-1, cbc=a15b3 >

2C2
20C2
20C2
10C22
10C22
20C22
20C22
2C10
4D5
4D5
2C8
5C23
5C23
10C8
10D4
10D4
2D10
2D10
4D10
4D10
5M4(2)
5C2×D4
2C52C8
2C40
2D20
2D20
5C4.D4

Smallest permutation representation of C20.46D4
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 36 16 21 11 26 6 31)(2 35 17 40 12 25 7 30)(3 34 18 39 13 24 8 29)(4 33 19 38 14 23 9 28)(5 32 20 37 15 22 10 27)
(1 16)(2 15)(3 14)(4 13)(5 12)(6 11)(7 10)(8 9)(17 20)(18 19)(21 26)(22 25)(23 24)(27 40)(28 39)(29 38)(30 37)(31 36)(32 35)(33 34)

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

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

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

C20.46D4 is a maximal subgroup of
D5×C4.D4  D20.3D4  M4(2).21D10  D20.6D4  D44D20  M4(2)⋊D10  C8.21D20  C8.24D20  M4(2).31D10  D4.3D20  D4.4D20  D2018D4  M4(2).D10  D20.39D4  M4(2).15D10  C60.28D4  C60.29D4  M4(2)⋊D15
C20.46D4 is a maximal quotient of
C42.D10  C53(C23⋊C8)  C4.Dic20  C4.D40  M4(2)⋊Dic5  C60.28D4  C60.29D4  M4(2)⋊D15

31 conjugacy classes

class 1 2A2B2C2D4A4B5A5B8A8B8C8D10A10B10C10D20A20B20C20D20E20F40A···40H
order12222445588881010101020202020202040···40
size1122020222244202022442222444···4

31 irreducible representations

dim1111122222244
type++++++++++
imageC1C2C2C2C4D4D5D10D20C5⋊D4C4×D5C4.D4C20.46D4
kernelC20.46D4C4.Dic5C5×M4(2)C2×D20C22×D5C20M4(2)C2×C4C4C4C22C5C1
# reps1111422244414

Matrix representation of C20.46D4 in GL4(𝔽41) generated by

143000
11900
001430
00119
,
30321923
27111322
525119
31361430
,
11900
143000
00119
001430
G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,14,11,0,0,30,9],[30,27,5,31,32,11,25,36,19,13,11,14,23,22,9,30],[11,14,0,0,9,30,0,0,0,0,11,14,0,0,9,30] >;

C20.46D4 in GAP, Magma, Sage, TeX

C_{20}._{46}D_4
% in TeX

G:=Group("C20.46D4");
// GroupNames label

G:=SmallGroup(160,30);
// by ID

G=gap.SmallGroup(160,30);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,86,297,4613]);
// Polycyclic

G:=Group<a,b,c|a^20=c^2=1,b^4=a^10,b*a*b^-1=c*a*c=a^-1,c*b*c=a^15*b^3>;
// generators/relations

Export

Subgroup lattice of C20.46D4 in TeX

׿
×
𝔽