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C5×D3×D399 #37561556

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

Aliases

direct productC5×D3×D399
unambiguous semidirect product(C3×D3)⋊(C5×D133)
unambiguous semidirect product(D3×C7)⋊(C5×D57)
unambiguous semidirect product(D3×C19)⋊(C5×D21)

Splittings

Fitting, socle, HallC32×C5×C7×C19 ⋊ C22
derived, HallC32×C7×C19 ⋊ C22×C5
HallC32×C5×C19 ⋊ C2×D7
HallC5×C7×C19 ⋊ D32
HallC32×C5×C7 ⋊ C2×D19
HallC32×C19 ⋊ C2×C5×D7
HallC7×C19 ⋊ C5×D32
HallC5×C19 ⋊ D3×D21
HallC32×C7 ⋊ C2×C5×D19
HallC32×C5 ⋊ C2×D133
HallC5×C7 ⋊ D3×D57
SylowC19 ⋊ C5×D3×D21
SylowC32 ⋊ C2×C5×D133
SylowC7 ⋊ C5×D3×D57

Subgroups

subgroupG/H
center, hypercenterC5D3×D399
commutator, cosocle, hypocenterC32×C7×C19C22×C5
FrattiniC1G
Fitting, socleC32×C5×C7×C19C22
LCSG ▷ C32×C7×C19
UCSC1 ◁ C5

176 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 3990

A minimal faithful permutation representation — 2 generators on 37 points:

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

Conjugacy classes & character table

3015 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 688);