Finite Group Databasebeta

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Cayley diagram

D5×#157 #67200

order360
isoclinismisoclinic class of D5×#157
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productD5×#157

Splittings

Fitting, socle, derived, HallC32×C5 ⋊ C2×C4
SylowC32 ⋊ C4×D5
SylowC5 ⋊ C2×#157

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C5C2×C4
prime radicalD5×C3⋊D3C2
cosocleC5×C3⋊D3C22
LCSG ▷ C32×C5
UCSC1

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2)(3,5,6,4)(7,8,10,11,9);(1,3,6)(2,5,4)(8,9)(10,11)

Conjugacy classes & character table

24 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽5)×#157, #10624306, #10624307, …
maximal normal subgroup ofAGL(1,𝔽5)×#157, #10624306, #10624307, …
maximal quotient ofDic5×#157, #10623985, #10623986, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 130);