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C2×C5×#16838465 #50304262

order38220
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2

Aliases

direct productC2×C5×#16838465

Splittings

Fitting, socleC2×C5×C72×C13 ⋊ C2×C3
derived, HallC72×C13 ⋊ C22×C3×C5
HallC2×C5×C7×C13×D7 ⋊ C3
HallC5×C13⋊#3862 ⋊ C22
HallC5×C72×C13 ⋊ C22×C3
HallC2×C7×C13×D7 ⋊ C3×C5
HallC13⋊#3862 ⋊ C22×C5
HallC2×C5×C7×D7 ⋊ C13⋊C3
HallC5×C72 ⋊ C22×C13⋊C3
HallC2×C7×D7 ⋊ C5×C13⋊C3
HallC5×C13 ⋊ C2×#64422
SylowC72 ⋊ C22×C5×C13⋊C3
SylowC13 ⋊ C2×C5×#64422

Subgroups

subgroupG/H
center, hypercenterC2×C5#16838465
commutator, cosocle, hypocenterC72×C13C22×C3×C5
FrattiniC1G
Fitting, socleC2×C5×C72×C13C2×C3
LCSG ▷ C72×C13
UCSC1 ◁ C2×C5

168 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 5460

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

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

Conjugacy classes & character table

1570 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 948);