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ΓL(1,𝔽8)×Dic391 #45659255

order32844
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic391
unambiguous semidirect productC23⋊(ΓL(1,𝔽8)×Dic17)
unambiguous semidirect product(C7×Dic391)⋊C3

Splittings

derived, HallC7×C17×C23 ⋊ C3×C4
HallC7×Dic391 ⋊ C3
HallC17×ΓL(1,𝔽8)×C23 ⋊ C4
HallΓL(1,𝔽8)×C23 ⋊ Dic17
HallC17×C23 ⋊ C4×ΓL(1,𝔽8)
HallC17×ΓL(1,𝔽8) ⋊ Dic23
HallC7×C23 ⋊ C3×Dic17
HallC7×C17 ⋊ C3×Dic23
SylowC23 ⋊ ΓL(1,𝔽8)×Dic17
SylowC17 ⋊ ΓL(1,𝔽8)×Dic23
SylowC7 ⋊ C3×Dic391

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D391
commutator, hypocenterC7×C17×C23C3×C4
Fitting, socle, cosocleC2×C7×C17×C23C2×C3
prime radicalC2×C17×ΓL(1,𝔽8)×C23C2
LCSG ▷ C7×C17×C23
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 10948

Conjugacy classes & character table

1970 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32844, 20);