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G = D204C4order 160 = 25·5

1st semidirect product of D20 and C4 acting via C4/C2=C2

metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D204C4, C423D5, C4.17D20, C20.33D4, Dic104C4, C53C4≀C2, (C4×C20)⋊6C2, C4.6(C4×D5), C20.37(C2×C4), C4○D20.1C2, (C2×C10).26D4, (C2×C4).66D10, C4.Dic51C2, (C2×C20).96C22, C22.7(C5⋊D4), C2.3(D10⋊C4), C10.12(C22⋊C4), SmallGroup(160,12)

Series: Derived Chief Lower central Upper central

C1C20 — D204C4
C1C5C10C2×C10C2×C20C4○D20 — D204C4
C5C10C20 — D204C4
C1C4C2×C4C42

Generators and relations for D204C4
 G = < a,b,c | a20=b2=c4=1, bab=a-1, ac=ca, cbc-1=a15b >

2C2
20C2
2C4
2C4
10C22
10C4
2C10
4D5
2C2×C4
5D4
5Q8
10C8
10D4
10C2×C4
2C20
2C20
2D10
2Dic5
5M4(2)
5C4○D4
2C5⋊D4
2C4×D5
2C52C8
2C2×C20
5C4≀C2

Smallest permutation representation of D204C4
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 40)(2 39)(3 38)(4 37)(5 36)(6 35)(7 34)(8 33)(9 32)(10 31)(11 30)(12 29)(13 28)(14 27)(15 26)(16 25)(17 24)(18 23)(19 22)(20 21)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26 31 36)(22 27 32 37)(23 28 33 38)(24 29 34 39)(25 30 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), (1,40)(2,39)(3,38)(4,37)(5,36)(6,35)(7,34)(8,33)(9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20)(21,26,31,36)(22,27,32,37)(23,28,33,38)(24,29,34,39)(25,30,35,40)>;

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

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

D204C4 is a maximal subgroup of
D4017C4  D4010C4  D5×C4≀C2  C42⋊D10  D44D20  M4(2).22D10  C42.196D10  M4(2)⋊D10  D4.9D20  D4.10D20  C424D10  C425D10  D20.14D4  D205D4  D20.15D4  C60.97D4  D6013C4  D607C4  C42⋊D15
D204C4 is a maximal quotient of
(C2×D20)⋊C4  C4⋊Dic5⋊C4  Dic103C8  D203C8  C42.D10  C42.2D10  C426Dic5  C60.97D4  D6013C4  D607C4

46 conjugacy classes

class 1 2A2B2C4A4B4C···4G4H5A5B8A8B10A···10F20A···20X
order1222444···44558810···1020···20
size11220112···2202220202···22···2

46 irreducible representations

dim111111222222222
type+++++++++
imageC1C2C2C2C4C4D4D4D5D10C4≀C2C4×D5D20C5⋊D4D204C4
kernelD204C4C4.Dic5C4×C20C4○D20Dic10D20C20C2×C10C42C2×C4C5C4C4C22C1
# reps1111221122444416

Matrix representation of D204C4 in GL2(𝔽41) generated by

50
033
,
033
50
,
400
032
G:=sub<GL(2,GF(41))| [5,0,0,33],[0,5,33,0],[40,0,0,32] >;

D204C4 in GAP, Magma, Sage, TeX

D_{20}\rtimes_4C_4
% in TeX

G:=Group("D20:4C4");
// GroupNames label

G:=SmallGroup(160,12);
// by ID

G=gap.SmallGroup(160,12);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,121,31,362,579,69,4613]);
// Polycyclic

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

Export

Subgroup lattice of D204C4 in TeX

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