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ΓL(1,𝔽8)×(C11⋊C5)⋊Dic5 #37528334

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C5 · Schur multiplier (H2) = C1

Aliases

direct productΓL(1,𝔽8)×(C11⋊C5)⋊Dic5
unambiguous semidirect product(C11⋊C5)⋊(Dic5×ΓL(1,𝔽8))
unambiguous semidirect productDic55⋊(C5×ΓL(1,𝔽8))
unambiguous semidirect product(ΓL(1,𝔽8)×Dic55)⋊C5
unambiguous semidirect product(C7×(C11⋊C5)⋊Dic5)⋊C3

Splittings

derivedC5×C7×C11 ⋊ C3×C4×C5
HallC7×(C11⋊C5)⋊Dic5 ⋊ C3
HallC5×ΓL(1,𝔽8)×C11⋊C5 ⋊ C4
HallC5×C7×C11⋊C5 ⋊ C3×C4
HallC5×C11⋊C5 ⋊ C4×ΓL(1,𝔽8)
HallC11×ΓL(1,𝔽8) ⋊ C5×Dic5
HallC7×C11 ⋊ C3×C5×Dic5
SylowC11 ⋊ C5×Dic5×ΓL(1,𝔽8)
SylowC7 ⋊ C3×(C11⋊C5)⋊Dic5

Subgroups

subgroupG/H
center, Frattini, hypercenterC2ΓL(1,𝔽8)×C5⋊AGL(1,𝔽11)
commutator, hypocenterC5×C7×C11C3×C4×C5
Fitting, socle, cosocleC2×C5×C7×C11C2×C3×C5
prime radicalC2×C5×ΓL(1,𝔽8)×C11⋊C5C2
LCSG ▷ C5×C7×C11
UCSC1 ◁ C2

128 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 1540

Conjugacy classes & character table

250 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 145);