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C5×C11×ΓL(1,𝔽8)⋊Dic5 #37528355

order23100
automorphism?
homologyAbelianization (H1) = C3×C4×C5×C11 · Schur multiplier (H2) = C1

Aliases

direct productC5×C11×ΓL(1,𝔽8)⋊Dic5
unambiguous semidirect productΓL(1,𝔽8)⋊(C5×C11×Dic5)
unambiguous semidirect productDic35⋊(C3×C5×C11)
unambiguous semidirect product(C5×Dic35)⋊(C3×C11)
unambiguous semidirect product(C5×C11×Dic35)⋊C3

Splittings

derivedC5×C7 ⋊ C3×C4×C5×C11
HallC5×C11×Dic35 ⋊ C3
HallC52×C11×ΓL(1,𝔽8) ⋊ C4
HallC52×C7×C11 ⋊ C3×C4
HallC5×Dic35 ⋊ C3×C11
HallC52×ΓL(1,𝔽8) ⋊ C4×C11
HallC52×C11 ⋊ ΓL(1,𝔽8)⋊C4
HallC11×ΓL(1,𝔽8) ⋊ C5×Dic5
HallC52×C7 ⋊ C3×C4×C11
HallC7×C11 ⋊ C3×C5×Dic5
SylowC52 ⋊ C11×ΓL(1,𝔽8)⋊C4
HallΓL(1,𝔽8) ⋊ C5×C11×Dic5
SylowC7 ⋊ C3×C5×C11×Dic5

Subgroups

subgroupG/H
center, hypercenterC2×C5×C11C5⋊AGL(1,𝔽7)
commutator, hypocenterC5×C7C3×C4×C5×C11
FrattiniC2C5×C11×C5⋊AGL(1,𝔽7)
Fitting, socleC2×C52×C7×C11C2×C3
prime radicalC2×C52×C11×ΓL(1,𝔽8)C2
cosocleC2×C5×C7C2×C3×C5×C11
LCSG ▷ C5×C7
UCSC1 ◁ C2×C5×C11

128 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 56 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

1870 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 166);