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ΓL(1,𝔽8)⋊C4×C113 #23748773

order9492
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C113 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)⋊C4×C113
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C113)
unambiguous semidirect productDic7⋊(C3×C113)
unambiguous semidirect product(Dic7×C113)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C113
HallDic7×C113 ⋊ C3
HallΓL(1,𝔽8)×C113 ⋊ C4
HallC7×C113 ⋊ C3×C4
HallDic7 ⋊ C3×C113
HallΓL(1,𝔽8) ⋊ C4×C113

Subgroups

subgroupG/H
center, hypercenterC2×C113AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C113
FrattiniC2AGL(1,𝔽7)×C113
Fitting, socleC2×C7×C113C2×C3
prime radicalC2×ΓL(1,𝔽8)×C113C2
cosocleC2×C7C2×C3×C113
LCSG ▷ C7
UCSC1 ◁ C2×C113

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1582 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofC3×ΓL(1,𝔽8)⋊C4×C113, C113×C3⋊(ΓL(1,𝔽8)⋊C4)
maximal normal subgroup ofC3×ΓL(1,𝔽8)⋊C4×C113
maximal quotient ofC113×Dic7⋊C9, C3×ΓL(1,𝔽8)⋊C4×C113, C113×C3⋊(ΓL(1,𝔽8)⋊C4)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9492, 15);