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C5×D13⋊C3×C7⋊D7 #50303807

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

Aliases

direct productC5×D13⋊C3×C7⋊D7
unambiguous semidirect productD13⋊(C3×C5×C7⋊D7)
unambiguous semidirect product(C7×D13⋊C3)⋊(C5×D7)

Splittings

Fitting, socle, HallC5×C72×C13 ⋊ C22×C3
derived, HallC72×C13 ⋊ C22×C3×C5
HallC5×D13×C7⋊D7 ⋊ C3
HallC5×C72×C13⋊C3 ⋊ C22
HallD13×C7⋊D7 ⋊ C3×C5
HallC72×C13⋊C3 ⋊ C22×C5
HallC5×C72 ⋊ C2×D13⋊C3
HallC5×C13⋊C3 ⋊ C2×C7⋊D7
HallC5×C13 ⋊ C2×C3×C7⋊D7
SylowC72 ⋊ C2×C5×D13⋊C3
HallC13⋊C3 ⋊ C2×C5×C7⋊D7
SylowC13 ⋊ C2×C3×C5×C7⋊D7

Subgroups

subgroupG/H
center, hypercenterC5D13⋊C3×C7⋊D7
commutator, cosocle, hypocenterC72×C13C22×C3×C5
FrattiniC1G
Fitting, socleC5×C72×C13C22×C3
LCSG ▷ C72×C13
UCSC1 ◁ C5

400 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 32
transitive degreeμt(G) = 3185

A minimal faithful permutation representation — 3 generators on 32 points:

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

Conjugacy classes & character table

1040 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(38220, 493);