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C5×C71×ΓL(1,𝔽8)⋊C4 #44447152

order29820
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C71 · Schur multiplier (H2) = C1

Aliases

direct productC5×C71×ΓL(1,𝔽8)⋊C4
unambiguous semidirect productΓL(1,𝔽8)⋊(C4×C5×C71)
unambiguous semidirect productDic7⋊(C3×C5×C71)
unambiguous semidirect product(C5×Dic7)⋊(C3×C71)
unambiguous semidirect product(C5×Dic7×C71)⋊C3

Splittings

derived, SylowC7 ⋊ C3×C4×C5×C71
HallC5×Dic7×C71 ⋊ C3
HallC5×ΓL(1,𝔽8)×C71 ⋊ C4
HallC5×C7×C71 ⋊ C3×C4
HallDic7×C71 ⋊ C3×C5
HallΓL(1,𝔽8)×C71 ⋊ C4×C5
HallC7×C71 ⋊ C3×C4×C5
HallC5×Dic7 ⋊ C3×C71
HallC5×ΓL(1,𝔽8) ⋊ C4×C71
HallC5×C7 ⋊ C3×C4×C71
HallDic7 ⋊ C3×C5×C71
HallΓL(1,𝔽8) ⋊ C4×C5×C71

Subgroups

subgroupG/H
center, hypercenterC2×C5×C71AGL(1,𝔽7)
commutator, hypocenterC7C3×C4×C5×C71
FrattiniC2C5×AGL(1,𝔽7)×C71
Fitting, socleC2×C5×C7×C71C2×C3
prime radicalC2×C5×ΓL(1,𝔽8)×C71C2
cosocleC2×C7C2×C3×C5×C71
LCSG ▷ C7
UCSC1 ◁ C2×C5×C71

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 9940

Conjugacy classes & character table

4970 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 40);