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C7×C53×ΓL(1,𝔽8)⋊C4 #44528350

order31164
automorphism?
homologyAbelianization (H1) = C3×C4×C7×C53 · Schur multiplier (H2) = C1

Aliases

direct productC7×C53×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C7×C53)
unambiguous semidirect productDic7⋊(C3×C7×C53)
unambiguous semidirect product(Dic7×C53)⋊(C3×C7)

Splittings

derivedC7 ⋊ C3×C4×C7×C53
HallC7×Dic7×C53 ⋊ C3
HallC7×ΓL(1,𝔽8)×C53 ⋊ C4
HallC72×C53 ⋊ C3×C4
HallC7×Dic7 ⋊ C3×C53
HallC7×ΓL(1,𝔽8) ⋊ C4×C53
SylowC72 ⋊ C3×C4×C53

Subgroups

subgroupG/H
center, hypercenterC2×C7×C53AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C7×C53
FrattiniC2C7×AGL(1,𝔽7)×C53
Fitting, socleC2×C72×C53C2×C3
prime radicalC2×C7×ΓL(1,𝔽8)×C53C2
cosocleC2×C7C2×C3×C7×C53
LCSG ▷ C7
UCSC1 ◁ C2×C7×C53

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 4452

Conjugacy classes & character table

5194 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(31164, 42);