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ΓL(1,𝔽8)×Dic355 #44447155

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

Aliases

direct productΓL(1,𝔽8)×Dic355
unambiguous semidirect productC71⋊(Dic5×ΓL(1,𝔽8))
unambiguous semidirect product(C7×Dic355)⋊C3

Splittings

derived, HallC5×C7×C71 ⋊ C3×C4
HallC7×Dic355 ⋊ C3
HallC5×ΓL(1,𝔽8)×C71 ⋊ C4
HallΓL(1,𝔽8)×C71 ⋊ Dic5
HallC7×C71 ⋊ C3×Dic5
HallC5×C71 ⋊ C4×ΓL(1,𝔽8)
HallC5×ΓL(1,𝔽8) ⋊ Dic71
SylowC71 ⋊ Dic5×ΓL(1,𝔽8)
HallC5×C7 ⋊ C3×Dic71
SylowC7 ⋊ C3×Dic355
SylowC5 ⋊ ΓL(1,𝔽8)×Dic71

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D355
commutator, hypocenterC5×C7×C71C3×C4
Fitting, socle, cosocleC2×C5×C7×C71C2×C3
prime radicalC2×C5×ΓL(1,𝔽8)×C71C2
LCSG ▷ C5×C7×C71
UCSC1 ◁ C2

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 9940

Conjugacy classes & character table

1790 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(29820, 43);