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ΓL(1,𝔽8)⋊C4×C109 #22603874

order9156
isoclinismisoclinic class of AGL(1,𝔽7)
automorphism?
homologyAbelianization (H1) = C3×C4×C109 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)⋊C4×C109
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C109)
unambiguous semidirect productDic7⋊(C3×C109)

Splittings

derived, SylowC7 ⋊ C3×C4×C109
HallDic7×C109 ⋊ C3
HallΓL(1,𝔽8)×C109 ⋊ C4
HallC7×C109 ⋊ C3×C4
HallDic7 ⋊ C3×C109
HallΓL(1,𝔽8) ⋊ C4×C109

Subgroups

subgroupG/H
center, hypercenterC2×C109AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C109
FrattiniC2AGL(1,𝔽7)×C109
Fitting, socleC2×C7×C109C2×C3
LCSG ▷ C7
UCSC1 ◁ C2×C109

24 conjugacy classes of subgroups.

Full subgroup data →

Conjugacy classes & character table

1526 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup ofΓL(1,𝔽8)⋊C4×C109⋊C3, C3×ΓL(1,𝔽8)⋊C4×C109, C109×C3⋊(ΓL(1,𝔽8)⋊C4)
maximal normal subgroup ofΓL(1,𝔽8)⋊C4×C109⋊C3, C3×ΓL(1,𝔽8)⋊C4×C109
maximal quotient ofC109×Dic7⋊C9, C3×ΓL(1,𝔽8)⋊C4×C109, C109×C3⋊(ΓL(1,𝔽8)⋊C4)

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(9156, 18);