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ΓL(1,𝔽8)⋊AGL(1,𝔽23) #24922328

order10626
automorphism?
homologyAbelianization (H1) = C2×C3×C11 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productΓL(1,𝔽8)⋊AGL(1,𝔽23)
unambiguous semidirect product(C23⋊C11)⋊AGL(1,𝔽7)
unambiguous semidirect product(ΓL(1,𝔽8)⋊D23)⋊C11
unambiguous semidirect product(C7⋊AGL(1,𝔽23))⋊C3

Splittings

Fitting, socle, derived, HallC7×C23 ⋊ C2×C3×C11
HallΓL(1,𝔽8)×C23⋊C11 ⋊ C2
HallC7⋊AGL(1,𝔽23) ⋊ C3
HallC7×C23⋊C11 ⋊ C2×C3
HallΓL(1,𝔽8)⋊D23 ⋊ C11
HallΓL(1,𝔽8)×C23 ⋊ C2×C11
HallD161 ⋊ C3×C11
HallC23⋊C11 ⋊ AGL(1,𝔽7)
SylowC23 ⋊ C11×AGL(1,𝔽7)
HallΓL(1,𝔽8) ⋊ AGL(1,𝔽23)
SylowC7 ⋊ C3×AGL(1,𝔽23)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC7×C23C2×C3×C11
LCSG ▷ C7×C23
UCSC1

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3,7,11,23|c^a=c^6,e^a=e^22,c^b=c^2,e^d=e^2

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofAGL(1,𝔽7)×AGL(1,𝔽23), C2×ΓL(1,𝔽8)⋊AGL(1,𝔽23), C3×ΓL(1,𝔽8)⋊AGL(1,𝔽23), …
maximal normal subgroup ofAGL(1,𝔽7)×AGL(1,𝔽23), C2×ΓL(1,𝔽8)⋊AGL(1,𝔽23), C3×ΓL(1,𝔽8)⋊AGL(1,𝔽23)
maximal quotient ofΓL(1,𝔽8)⋊(Dic23⋊C11), C2×ΓL(1,𝔽8)⋊AGL(1,𝔽23), (C7⋊C9)⋊AGL(1,𝔽23), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(10626, 36);