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C5×D11×ΓL(1,𝔽8)⋊D7 #44568627

order32340
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D11×ΓL(1,𝔽8)⋊D7

Splittings

Fitting, socle, HallC5×C72×C11 ⋊ C22×C3
derived, HallC72×C11 ⋊ C22×C3×C5
HallC5×D11×C7⋊D7 ⋊ C3
HallC5×C7×C11×ΓL(1,𝔽8) ⋊ C22
HallD11×C7⋊D7 ⋊ C3×C5
HallC7×C11×ΓL(1,𝔽8) ⋊ C22×C5
HallC5×C7×ΓL(1,𝔽8) ⋊ C2×D11
HallC5×C72 ⋊ C2×C3×D11
HallC7×ΓL(1,𝔽8) ⋊ C2×C5×D11
HallC5×C11 ⋊ C2×ΓL(1,𝔽8)⋊D7
SylowC72 ⋊ C2×C3×C5×D11
SylowC11 ⋊ C2×C5×ΓL(1,𝔽8)⋊D7

Subgroups

subgroupG/H
center, hypercenterC5D11×ΓL(1,𝔽8)⋊D7
commutator, cosocle, hypocenterC72×C11C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C11C22×C3
LCSG ▷ C72×C11
UCSC1 ◁ C5

200 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degreeμ(G) = 30
transitive degreeμt(G) = 1155

A minimal faithful permutation representation — 2 generators on 30 points:

(1,2,4,6,8,10,11,9,7,5,3)(13,14)(15,16)(17,18)(20,23,24,21,22,25)(26,27,28,29,30);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13,15,17,18,16,14)(19,20,22,25,24,23,21)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32340, 329);