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ΓL(1,𝔽8)⋊C4×C389 #45658069

order32676
automorphism?
homologyAbelianization (H1) = C3×C4×C389 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)⋊C4×C389
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C389)
unambiguous semidirect productDic7⋊(C3×C389)
unambiguous semidirect product(Dic7×C389)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C389
HallDic7×C389 ⋊ C3
HallΓL(1,𝔽8)×C389 ⋊ C4
HallC7×C389 ⋊ C3×C4
HallDic7 ⋊ C3×C389
HallΓL(1,𝔽8) ⋊ C4×C389

Subgroups

subgroupG/H
center, hypercenterC2×C389AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C389
FrattiniC2AGL(1,𝔽7)×C389
Fitting, socleC2×C7×C389C2×C3
prime radicalC2×ΓL(1,𝔽8)×C389C2
cosocleC2×C7C2×C3×C389
LCSG ▷ C7
UCSC1 ◁ C2×C389

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 10892

Conjugacy classes & character table

5446 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32676, 11);