Finite Group Databasebeta

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C2×C5×C3⋊(C11⋊(D13⋊C3)) #38788136

order25740
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×C3⋊(C11⋊(D13⋊C3))
unambiguous semidirect product(C2×C3)⋊(C5×C11⋊(D13⋊C3))
unambiguous semidirect product(C2×C13⋊C3)⋊(C5×D33)
unambiguous semidirect product(C2×C5×D429)⋊C3

Splittings

Fitting, socleC2×C3×C5×C11×C13 ⋊ C2×C3
derivedC3×C11×C13 ⋊ C22×C3×C5
HallC3×C5×C11×C13⋊C3 ⋊ C22
HallC3×C11×C13⋊C3 ⋊ C22×C5
HallC5×C11×C13 ⋊ C2×C3×D3
HallC3×C5×C13⋊C3 ⋊ C2×D11
HallC11×C13 ⋊ C2×C3×C5×D3
HallC3×C13⋊C3 ⋊ C2×C5×D11
HallC5×C13 ⋊ C2×C3×D33
HallC5×C11 ⋊ C2×C3⋊(D13⋊C3)
SylowC13 ⋊ C2×C3×C5×D33
SylowC11 ⋊ C2×C5×C3⋊(D13⋊C3)

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊(C11⋊(D13⋊C3))
commutator, cosocle, hypocenterC3×C11×C13C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C11×C13C2×C3
LCSG ▷ C3×C11×C13
UCSC1 ◁ C2×C5

176 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 34
transitive degreeμt(G) = 4290

A minimal faithful permutation representation — 2 generators on 34 points:

(2,6,7,3,4,8)(5,12,11,9,10,13)(15,16)(17,18)(19,20)(21,22)(23,24)(26,27)(28,29,30,31,32);(1,2,5,4,8,11,10,12,13,7,6,9,3)(14,15,17,19,21,23,24,22,20,18,16)(25,26,27)(33,34)

Conjugacy classes & character table

1200 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(25740, 367);