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C11×C23×ΓL(1,𝔽8)⋊C4 #36318352

order21252
automorphism?
homologyAbelianization (H1) = C3×C4×C11×C23 · Schur multiplier (H2) = C1

Aliases

direct productC11×C23×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C11×C23)
unambiguous semidirect productDic7⋊(C3×C11×C23)
unambiguous semidirect product(C11×Dic7)⋊(C3×C23)
unambiguous semidirect product(C11×C23×Dic7)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C11×C23
HallC11×C23×Dic7 ⋊ C3
HallC11×ΓL(1,𝔽8)×C23 ⋊ C4
HallC7×C11×C23 ⋊ C3×C4
HallC23×Dic7 ⋊ C3×C11
HallΓL(1,𝔽8)×C23 ⋊ C4×C11
HallC11×Dic7 ⋊ C3×C23
HallC11×ΓL(1,𝔽8) ⋊ C4×C23
HallC7×C23 ⋊ C3×C4×C11
HallC7×C11 ⋊ C3×C4×C23
HallDic7 ⋊ C3×C11×C23
HallΓL(1,𝔽8) ⋊ C4×C11×C23

Subgroups

subgroupG/H
center, hypercenterC2×C11×C23AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C11×C23
FrattiniC2C11×C23×AGL(1,𝔽7)
Fitting, socleC2×C7×C11×C23C2×C3
prime radicalC2×C11×ΓL(1,𝔽8)×C23C2
cosocleC2×C7C2×C3×C11×C23
LCSG ▷ C7
UCSC1 ◁ C2×C11×C23

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7084

Conjugacy classes & character table

3542 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(21252, 30);