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C22×C3×ΓL(1,𝔽8)×C23 #19154010

order5796
isoclinismisoclinic class of ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C22×C32×C23 · Schur multiplier (H2) = C2×C3

Aliases

direct productC22×C3×ΓL(1,𝔽8)×C23

Splittings

Fitting, socleC22×C3×C7×C23 ⋊ C3
derived, SylowC7 ⋊ C22×C32×C23
HallC22×C7×C23 ⋊ C32
HallC7×C23 ⋊ C22×C32
HallC22×C7 ⋊ C32×C23

Subgroups

subgroupG/H
center, hypercenterC22×C3×C23ΓL(1,𝔽8)
commutator, hypocenterC7C22×C32×C23
FrattiniC1G
Fitting, socleC22×C3×C7×C23C3
LCSG ▷ C7
UCSC1 ◁ C22×C3×C23

120 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1380 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC2×C3×D5×ΓL(1,𝔽8)×C23, C22×C3×C5×ΓL(1,𝔽8)×C23
maximal normal subgroup ofC22×C3×C5×ΓL(1,𝔽8)×C23
maximal quotient ofC2×C3×D5×ΓL(1,𝔽8)×C23, C22×C3×C5×ΓL(1,𝔽8)×C23

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(5796, 100);