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C7×D5×D165 #37528675

order23100
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2

Aliases

direct productC7×D5×D165
unambiguous semidirect product(C3×D5)⋊(C7×D55)
unambiguous semidirect product(C5×D5)⋊(C7×D33)
unambiguous semidirect product(D5×C11)⋊(C7×D15)

Splittings

Fitting, socle, HallC3×C52×C7×C11 ⋊ C22
derived, HallC3×C52×C11 ⋊ C22×C7
HallC52×C7×C11 ⋊ C2×D3
HallC3×C52×C7 ⋊ C2×D11
HallC52×C11 ⋊ C2×D3×C7
HallC3×C7×C11 ⋊ D52
HallC52×C7 ⋊ C2×D33
HallC7×C11 ⋊ D5×D15
HallC3×C52 ⋊ C2×C7×D11
HallC3×C11 ⋊ C7×D52
SylowC52 ⋊ C2×C7×D33
HallC3×C7 ⋊ D5×D55
SylowC11 ⋊ C7×D5×D15
SylowC3 ⋊ C7×D5×D55

Subgroups

subgroupG/H
center, hypercenterC7D5×D165
commutator, cosocle, hypocenterC3×C52×C11C22×C7
FrattiniC1G
Fitting, socleC3×C52×C7×C11C22
LCSG ▷ C3×C52×C11
UCSC1 ◁ C7

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

normal subgroups only

This group has 192 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) = 31
transitive degreeμt(G) = 2310

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

2352 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(23100, 486);