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C2×C5×C3⋊((C19⋊C3)⋊D7) #37561479

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

Aliases

direct productC2×C5×C3⋊((C19⋊C3)⋊D7)
unambiguous semidirect product(C2×C3)⋊(C5×(C19⋊C3)⋊D7)
unambiguous semidirect product(C2×C19⋊C3)⋊(C5×D21)

Splittings

Fitting, socleC2×C3×C5×C7×C19 ⋊ C2×C3
derivedC3×C7×C19 ⋊ C22×C3×C5
HallC3×C5×C7×C19⋊C3 ⋊ C22
HallC3×C7×C19⋊C3 ⋊ C22×C5
HallC3×C5×C19⋊C3 ⋊ C2×D7
HallC5×C7×C19 ⋊ C2×C3×D3
HallC3×C19⋊C3 ⋊ C2×C5×D7
HallC7×C19 ⋊ C2×C3×C5×D3
HallC5×C19 ⋊ C2×C3×D21
HallC5×C7 ⋊ C2×C3⋊(D19⋊C3)
SylowC19 ⋊ C2×C3×C5×D21
SylowC7 ⋊ C2×C5×C3⋊(D19⋊C3)

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊((C19⋊C3)⋊D7)
commutator, cosocle, hypocenterC3×C7×C19C22×C3×C5
FrattiniC1G
Fitting, socleC2×C3×C5×C7×C19C2×C3
LCSG ▷ C3×C7×C19
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) = 36
transitive degreeμt(G) = 3990

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 611);