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Cayley diagram

D100 #6019

order200
isoclinismisoclinic class of Dic50
automorphismAut(G) = D4×AGL(1,ℤ/25) (order 4000) · Inn(G) = C2×D25 · Out(G) = C2×C4×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

dihedralD100
unambiguous semidirect productC4⋊D25

Splittings

FittingC4×C25 ⋊ C2
SylowC25 ⋊ D4

Subgroups

subgroupG/H
centerC2C2×D25
commutator, cosocleC2×C25C22
Frattini, socleC2×C5C2×D5
FittingC4×C25C2
hypercenterC4D25
hypocenterC25D4
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ C4

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 100

A minimal faithful permutation representation — 2 generators on 29 points:

(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);(2,4)(6,29)(7,28)(8,27)(9,26)(10,25)(11,24)(12,23)(13,22)(14,21)(15,20)(16,19)(17,18)

Polycyclic presentation:

2^3,5^2|b^2=c,d^5=e^4,b^a=bc,d^a=d^4e,e^a=e^4

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#87713, D200, #87722, …
maximal normal subgroup of#87713, D200, #87722, …
maximal quotient ofDic100, #87713, D200, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 6);