Finite Group Databasebeta

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C5×D3×C11⋊(D19⋊C3) #49178586

order37620
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC5×D3×C11⋊(D19⋊C3)
unambiguous semidirect product(D3×C19⋊C3)⋊(C5×D11)
unambiguous semidirect product(C5×D3×D209)⋊C3

Splittings

Fitting, socleC3×C5×C11×C19 ⋊ C22×C3
derivedC3×C11×C19 ⋊ C22×C3×C5
HallC3×C5×C11×C19⋊C3 ⋊ C22
HallC3×C11×C19⋊C3 ⋊ C22×C5
HallC5×C11×C19 ⋊ C2×C3×D3
HallC3×C5×C19⋊C3 ⋊ C2×D11
HallC11×C19 ⋊ C2×C3×C5×D3
HallC3×C19⋊C3 ⋊ C2×C5×D11
HallC5×C19 ⋊ C3×D3×D11
HallC5×C11 ⋊ D3×D19⋊C3
SylowC19 ⋊ C3×C5×D3×D11
SylowC11 ⋊ C5×D3×D19⋊C3

Subgroups

subgroupG/H
center, hypercenterC5D3×C11⋊(D19⋊C3)
commutator, cosocle, hypocenterC3×C11×C19C22×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C11×C19C22×C3
LCSG ▷ C3×C11×C19
UCSC1 ◁ C5

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

normal subgroups only

This group has 176 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) = 38
transitive degreeμt(G) = 3135

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3)(4,5,6,7,8)(10,11)(12,13)(14,15)(16,17)(18,19)(21,25,26,22,23,27)(24,32,33,28,29,37)(30,38,36,35,34,31);(2,3)(9,10,12,14,16,18,19,17,15,13,11)(20,21,24,29,34,37,36,26,25,30,35,23,27,31,33,38,32,28,22)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 295);