Finite Group Databasebeta

← Prev (#11929477) Next (#11929479) →

C22×C5×C7⋊C9 #11929478

order1260
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C5×C9 · Schur multiplier (H2) = C2

Aliases

direct productC22×C5×C7⋊C9

Splittings

derived, SylowC7 ⋊ C22×C5×C9
HallC22×C5×C7 ⋊ C9
HallC5×C7 ⋊ C22×C9
HallC22×C7 ⋊ C5×C9

Subgroups

subgroupG/H
center, hypercenterC22×C3×C5ΓL(1,𝔽8)
commutator, hypocenterC7C22×C5×C9
FrattiniC3C22×C5×ΓL(1,𝔽8)
Fitting, socle, prime radicalC22×C3×C5×C7C3
cosocleC3×C7C22×C3×C5
LCSG ▷ C7
UCSC1 ◁ C22×C3×C5

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 25
transitive degreeμt(G) = 1260

A minimal faithful permutation representation — 2 generators on 25 points:

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

Conjugacy classes & character table

300 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C7⋊C9×AGL(1,𝔽11), C22×C11⋊C5×C7⋊C9, C2×C5×D11×C7⋊C9, …
maximal normal subgroup ofC22×C5×C11×C7⋊C9, C22×C5×C13×C7⋊C9, C22×C5×C17×C7⋊C9, …
maximal quotient ofC2×C7⋊C9×AGL(1,𝔽11), C22×C11⋊C5×C7⋊C9, C2×C5×D11×C7⋊C9, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1260, 30);