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C3⋊D3×C19⋊AGL(1,𝔽11) #49178559

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

Aliases

direct productC3⋊D3×C19⋊AGL(1,𝔽11)
unambiguous semidirect product(C3⋊D3×D209)⋊C5

Splittings

Fitting, socle, derived, HallC32×C11×C19 ⋊ C22×C5
HallC32×C19×C11⋊C5 ⋊ C22
HallC3⋊D3×D209 ⋊ C5
HallC19×C11⋊C5 ⋊ C2×C3⋊D3
HallC32×C11⋊C5 ⋊ C2×D19
HallC11×C19 ⋊ C2×C5×C3⋊D3
HallC32×C19 ⋊ C2×AGL(1,𝔽11)
HallC32×C11 ⋊ C2×C5×D19
HallC11⋊C5 ⋊ C3⋊D3×D19
SylowC19 ⋊ C3⋊D3×AGL(1,𝔽11)
SylowC11 ⋊ C5×C3⋊D3×D19
SylowC32 ⋊ C2×C19⋊AGL(1,𝔽11)

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC32×C11×C19C22×C5
LCSG ▷ C32×C11×C19
UCSC1

240 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 58 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) = 36
transitive degreeμt(G) = 1881

A minimal faithful permutation representation — 3 generators on 36 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37620, 268);