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C23×ΓL(1,𝔽8)⋊#261 #38737349

order25116
automorphism?
homologyAbelianization (H1) = C3×C4×C23 · Schur multiplier (H2) = C1

Aliases

direct productC23×ΓL(1,𝔽8)⋊#261
unambiguous semidirect productΓL(1,𝔽8)⋊(C23×#261)

Splittings

Fitting, socle, HallC7×C13×C23 ⋊ C3×C4
derived, HallC7×C13 ⋊ C3×C4×C23
HallC23×C7⋊#261 ⋊ C3
HallC13×ΓL(1,𝔽8)×C23 ⋊ C4
HallΓL(1,𝔽8)×C23 ⋊ #261
HallC7⋊#261 ⋊ C3×C23
HallC13×C23 ⋊ ΓL(1,𝔽8)⋊C4
HallC13×ΓL(1,𝔽8) ⋊ C4×C23
HallC7×C23 ⋊ C3×#261
HallΓL(1,𝔽8) ⋊ C23×#261
SylowC13 ⋊ C23×ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C3×C23×#261

Subgroups

subgroupG/H
center, hypercenterC23ΓL(1,𝔽8)⋊#261
commutator, hypocenterC7×C13C3×C4×C23
FrattiniC1G
Fitting, socleC7×C13×C23C3×C4
prime radicalΓL(1,𝔽8)×C23×D13C2
cosocleC7×D13C2×C3×C23
LCSG ▷ C7×C13
UCSC1 ◁ C23

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 2093

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25116, 68);