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C52×C11⋊(ΓL(1,𝔽8)⋊C4) #37528353

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

Aliases

direct productC52×C11⋊(ΓL(1,𝔽8)⋊C4)
unambiguous semidirect productΓL(1,𝔽8)⋊(C52×Dic11)

Splittings

derived, HallC7×C11 ⋊ C3×C4×C52
HallC52×Dic77 ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C4
HallC52×C7×C11 ⋊ C3×C4
HallC52×ΓL(1,𝔽8) ⋊ Dic11
HallDic77 ⋊ C3×C52
HallC52×C11 ⋊ ΓL(1,𝔽8)⋊C4
HallC11×ΓL(1,𝔽8) ⋊ C4×C52
HallC52×C7 ⋊ C3×Dic11
HallΓL(1,𝔽8) ⋊ C52×Dic11
SylowC11 ⋊ C52×ΓL(1,𝔽8)⋊C4
SylowC7 ⋊ C3×C52×Dic11

Subgroups

subgroupG/H
center, hypercenterC2×C52C11⋊AGL(1,𝔽7)
commutator, hypocenterC7×C11C3×C4×C52
FrattiniC2C52×C11⋊AGL(1,𝔽7)
Fitting, socleC2×C52×C7×C11C2×C3
prime radicalC2×C52×C11×ΓL(1,𝔽8)C2
cosocleC2×C7×C11C2×C3×C52
LCSG ▷ C7×C11
UCSC1 ◁ C2×C52

192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 7700

Conjugacy classes & character table

1600 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 164);