Finite Group Databasebeta

← Prev (#6020) Next (#6022) →

Cayley diagram

C22⋊D25 #6021

order200
isoclinismisoclinic class of Dic50
automorphismAut(G) = C22×AGL(1,ℤ/25) (order 2000) · Inn(G) = C2×D25 · Out(G) = C4×C5
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect productC22⋊D25

Splittings

FittingC22×C25 ⋊ C2
SylowC25 ⋊ D4

Subgroups

subgroupG/H
centerC2C2×D25
commutator, cosocleC2×C25C22
Frattini, socleC2×C5C2×D5
FittingC22×C25C2
hypercenterC22D25
hypocenterC25D4
LCSG ▷ C2×C25 ▷ C25
UCSC1 ◁ C2 ◁ C22

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:

(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(27,29);(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(26,27)(28,29)

Polycyclic presentation:

2^3,5^2|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#87744, D4×D25, #87746, …
maximal normal subgroup of#87744, D4×D25, #87746, …
maximal quotient of#87718, #87720, #87721, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(200, 8);