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C3×C5×D13×C7⋊D7 #50304346

order38220
automorphism?
homologyAbelianization (H1) = C22×C3×C5 · Schur multiplier (H2) = C2×C7

Aliases

direct productC3×C5×D13×C7⋊D7

Splittings

Fitting, socle, HallC3×C5×C72×C13 ⋊ C22
derived, HallC72×C13 ⋊ C22×C3×C5
HallC5×C72×C13 ⋊ C22×C3
HallC3×C72×C13 ⋊ C22×C5
HallC3×C5×C72 ⋊ C2×D13
HallC5×C72 ⋊ C2×C3×D13
HallC3×C5×C13 ⋊ C2×C7⋊D7
HallC3×C72 ⋊ C2×C5×D13
HallC5×C13 ⋊ C2×C3×C7⋊D7
SylowC72 ⋊ C2×C3×C5×D13
HallC3×C13 ⋊ C2×C5×C7⋊D7
SylowC13 ⋊ C2×C3×C5×C7⋊D7

Subgroups

subgroupG/H
center, hypercenterC3×C5D13×C7⋊D7
commutator, cosocle, hypocenterC72×C13C22×C3×C5
FrattiniC1G
Fitting, socleC3×C5×C72×C13C22
LCSG ▷ C72×C13
UCSC1 ◁ C3×C5

400 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 35
transitive degreeμt(G) = 9555

A minimal faithful permutation representation — 3 generators on 35 points:

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

Conjugacy classes & character table

3120 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 1032);