Finite Group Databasebeta

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C7×C3⋊D3×D65 #30655700

order16380
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C2×C3

Aliases

direct productC7×C3⋊D3×D65

Splittings

Fitting, socle, HallC32×C5×C7×C13 ⋊ C22
derived, HallC32×C5×C13 ⋊ C22×C7
HallC32×C7×C13 ⋊ C2×D5
HallC5×C7×C13 ⋊ C2×C3⋊D3
HallC32×C5×C7 ⋊ C2×D13
HallC32×C13 ⋊ C2×C7×D5
HallC7×C13 ⋊ D5×C3⋊D3
HallC5×C13 ⋊ C2×C7×C3⋊D3
HallC32×C7 ⋊ C2×D65
HallC32×C5 ⋊ C2×C7×D13
HallC5×C7 ⋊ C3⋊D3×D13
SylowC13 ⋊ C7×D5×C3⋊D3
SylowC32 ⋊ C2×C7×D65
SylowC5 ⋊ C7×C3⋊D3×D13

Subgroups

subgroupG/H
center, hypercenterC7C3⋊D3×D65
commutator, cosocle, hypocenterC32×C5×C13C22×C7
FrattiniC1G
Fitting, socleC32×C5×C7×C13C22
LCSG ▷ C32×C5×C13
UCSC1 ◁ C7

240 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 4095

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

1428 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16380, 750);