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ΓL(1,𝔽8)⋊C4×C269 #36378947

order22596
automorphism?
homologyAbelianization (H1) = C3×C4×C269 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)⋊C4×C269
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C269)
unambiguous semidirect productDic7⋊(C3×C269)
unambiguous semidirect product(Dic7×C269)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C269
HallDic7×C269 ⋊ C3
HallΓL(1,𝔽8)×C269 ⋊ C4
HallC7×C269 ⋊ C3×C4
HallDic7 ⋊ C3×C269
HallΓL(1,𝔽8) ⋊ C4×C269

Subgroups

subgroupG/H
center, hypercenterC2×C269AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C269
FrattiniC2AGL(1,𝔽7)×C269
Fitting, socleC2×C7×C269C2×C3
prime radicalC2×ΓL(1,𝔽8)×C269C2
cosocleC2×C7C2×C3×C269
LCSG ▷ C7
UCSC1 ◁ C2×C269

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7532

Conjugacy classes & character table

3766 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22596, 11);