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C5×ΓL(1,𝔽8)⋊(Dic11⋊C5) #37528329

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C52 · Schur multiplier (H2) = C5

Aliases

direct productC5×ΓL(1,𝔽8)⋊(Dic11⋊C5)
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×Dic11⋊C5)
unambiguous semidirect product(C11⋊C5)⋊(C5×ΓL(1,𝔽8)⋊C4)
unambiguous semidirect product(C11⋊(ΓL(1,𝔽8)⋊C4))⋊C52
unambiguous semidirect product(C7⋊(Dic11⋊C5))⋊(C3×C5)
unambiguous semidirect product(C5×C11⋊(ΓL(1,𝔽8)⋊C4))⋊C5
unambiguous semidirect product(C5×C7⋊(Dic11⋊C5))⋊C3

Splittings

derived, HallC7×C11 ⋊ C3×C4×C52
HallC5×C7⋊(Dic11⋊C5) ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC5×C7×C11⋊C5 ⋊ C3×C4
HallC11⋊(ΓL(1,𝔽8)⋊C4) ⋊ C52
HallDic77 ⋊ C3×C52
HallC5×C11⋊C5 ⋊ ΓL(1,𝔽8)⋊C4
HallC11×ΓL(1,𝔽8) ⋊ C4×C52
HallΓL(1,𝔽8) ⋊ C5×Dic11⋊C5
SylowC11 ⋊ C52×ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C3×C5×Dic11⋊C5

Subgroups

subgroupG/H
center, hypercenterC2×C5ΓL(1,𝔽8)⋊AGL(1,𝔽11)
commutator, hypocenterC7×C11C3×C4×C52
FrattiniC2C5×ΓL(1,𝔽8)⋊AGL(1,𝔽11)
Fitting, socleC2×C5×C7×C11C2×C3×C5
prime radicalC2×C5×ΓL(1,𝔽8)×C11⋊C5C2
cosocleC2×C7×C11C2×C3×C52
LCSG ▷ C7×C11
UCSC1 ◁ C2×C5

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 1540

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 140);