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C5×D133⋊(C3⋊D3) #37561593

order23940
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C2×C3

Aliases

direct productC5×D133⋊(C3⋊D3)
unambiguous semidirect productD133⋊(C5×C3⋊D3)

Splittings

Fitting, socle, HallC32×C5×C7×C19 ⋊ C22
derived, HallC32×C7×C19 ⋊ C22×C5
HallC32×C5×C19 ⋊ C2×D7
HallC5×C7×C19 ⋊ C2×C3⋊D3
HallC32×C5×C7 ⋊ C2×D19
HallC32×C19 ⋊ C2×C5×D7
HallC7×C19 ⋊ C2×C5×C3⋊D3
HallC5×C19 ⋊ D7×C3⋊D3
HallC32×C7 ⋊ C2×C5×D19
HallC32×C5 ⋊ D7×D19
HallC5×C7 ⋊ C3⋊D3×D19
SylowC19 ⋊ C5×D7×C3⋊D3
SylowC32 ⋊ C5×D7×D19
SylowC7 ⋊ C5×C3⋊D3×D19

Subgroups

subgroupG/H
center, hypercenterC5D133⋊(C3⋊D3)
commutator, cosocle, hypocenterC32×C7×C19C22×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C19C22
LCSG ▷ C32×C7×C19
UCSC1 ◁ C5

240 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 37
transitive degreeμt(G) = 5985

A minimal faithful permutation representation — 3 generators on 37 points:

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

Conjugacy classes & character table

1635 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 725);