Finite Group Databasebeta

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C5×#11716321⋊D3 #37561295

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

Aliases

direct productC5×#11716321⋊D3
unambiguous semidirect product#11716321⋊(C5×D3)

Splittings

Fitting, socleC3×C5×C7×C19 ⋊ C22×C3
derivedC3×C7×C19 ⋊ C22×C3×C5
HallC3×C5×#87704 ⋊ C22
HallC3×#87704 ⋊ C22×C5
HallC5×C7×C19 ⋊ C2×C3×D3
HallC7×C19 ⋊ C2×C3×C5×D3
HallC5×C19 ⋊ C2×C3⋊AGL(1,𝔽7)
HallC5×C7 ⋊ D3×D19⋊C3
SylowC19 ⋊ C2×C5×C3⋊AGL(1,𝔽7)
SylowC7 ⋊ C5×D3×D19⋊C3

Subgroups

subgroupG/H
center, hypercenterC5#11716321⋊D3
commutator, cosocle, hypocenterC3×C7×C19C22×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C7×C19C22×C3
LCSG ▷ C3×C7×C19
UCSC1 ◁ C5

176 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 34
transitive degreeμt(G) = 1995

A minimal faithful permutation representation — 2 generators on 34 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23940, 427);