Finite Group Databasebeta

← Prev (#14523666) Next (#14523668) →

C22×C52×ΓL(1,𝔽8) #14523667

order2100
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C3×C52 · Schur multiplier (H2) = C2×C5

Aliases

direct productC22×C52×ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC22×C52×C7 ⋊ C3
derived, SylowC7 ⋊ C22×C3×C52
HallC52×C7 ⋊ C22×C3
HallC22×C7 ⋊ C3×C52

Subgroups

subgroupG/H
center, hypercenterC22×C52ΓL(1,𝔽8)
commutator, cosocle, hypocenterC7C22×C3×C52
FrattiniC1G
Fitting, socleC22×C52×C7C3
LCSG ▷ C7
UCSC1 ◁ C22×C52

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 700

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

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

Relationships

maximal subgroup ofC22×C52×C49⋊C3, C2×C52×#64423, C2×C52×#64422, …
maximal normal subgroup ofC22×C52×C7×ΓL(1,𝔽8), C22×C52×C11×ΓL(1,𝔽8), C22×C52×C13×ΓL(1,𝔽8), …
maximal quotient ofC22×C52×C49⋊C3, C2×C52×#64423, C2×C52×#64422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2100, 147);