Finite Group Databasebeta

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C2×C5×C3⋊#11716328 #37561469

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

Aliases

direct productC2×C5×C3⋊#11716328
unambiguous semidirect product(C2×C3)⋊(C5×#11716328)

Splittings

Fitting, socleC2×C3×C5×C7×C19 ⋊ C2×C3
derivedC3×C7×C19 ⋊ C22×C3×C5
HallC3×C5×#87705 ⋊ C22
HallC3×#87705 ⋊ C22×C5
HallC5×C7×C19 ⋊ C2×C3×D3
HallC7×C19 ⋊ C2×C3×C5×D3
HallC5×C19 ⋊ C2×C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ C2×C3⋊(D19⋊C3)
SylowC19 ⋊ C2×C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C2×C5×C3⋊(D19⋊C3)

Subgroups

subgroupG/H
center, hypercenterC2×C5C3⋊#11716328
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 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) = 36
transitive degreeμt(G) = 3990

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

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 601);