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ΓL(1,𝔽8)×Dic345 #44394451

order28980
automorphism?
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×Dic345
unambiguous semidirect productC3⋊(ΓL(1,𝔽8)×Dic115)
unambiguous semidirect productC23⋊(ΓL(1,𝔽8)×Dic15)
unambiguous semidirect product(C7×Dic345)⋊C3

Splittings

derivedC3×C5×C7×C23 ⋊ C3×C4
HallC3×C5×ΓL(1,𝔽8)×C23 ⋊ C4
HallC3×ΓL(1,𝔽8)×C23 ⋊ Dic5
HallC5×C7×C23 ⋊ C3×Dic3
HallC3×C5×ΓL(1,𝔽8) ⋊ Dic23
HallC7×C23 ⋊ C3×Dic15
HallC5×C23 ⋊ Dic3×ΓL(1,𝔽8)
HallC3×ΓL(1,𝔽8) ⋊ Dic115
HallC5×C7 ⋊ C3×Dic69
SylowC23 ⋊ ΓL(1,𝔽8)×Dic15
SylowC7 ⋊ C3×Dic345
SylowC5 ⋊ ΓL(1,𝔽8)×Dic69

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×D345
commutator, hypocenterC3×C5×C7×C23C3×C4
Fitting, socle, cosocleC2×C3×C5×C7×C23C2×C3
prime radicalC2×C3×C5×ΓL(1,𝔽8)×C23C2
LCSG ▷ C3×C5×C7×C23
UCSC1 ◁ C2

112 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 112 conjugacy classes of subgroups — too many to draw. Only its 51 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 9660

Conjugacy classes & character table

1740 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(28980, 120);