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C3×C5⋊D5×D37⋊C3 #45686235

order33300
automorphism?
homologyAbelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3×C5

Aliases

direct productC3×C5⋊D5×D37⋊C3
unambiguous semidirect productD37⋊(C32×C5⋊D5)
unambiguous semidirect product(C5×D37⋊C3)⋊(C3×D5)

Splittings

Fitting, socleC3×C52×C37 ⋊ C22×C3
derived, HallC52×C37 ⋊ C22×C32
HallC3×C52×C37⋊C3 ⋊ C22
HallC5⋊D5×D37 ⋊ C32
HallC3×C37⋊C3 ⋊ C2×C5⋊D5
SylowC37 ⋊ C2×C32×C5⋊D5
SylowC52 ⋊ C2×C3×D37⋊C3

Subgroups

subgroupG/H
center, hypercenterC3C5⋊D5×D37⋊C3
commutator, cosocle, hypocenterC52×C37C22×C32
FrattiniC1G
Fitting, socleC3×C52×C37C22×C3
LCSG ▷ C52×C37
UCSC1 ◁ C3

480 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 50
transitive degreeμt(G) = 2775

A minimal faithful permutation representation — 3 generators on 50 points:

(9,10,12,13,11)(15,19,20,16,17,21)(18,27,28,22,23,36)(24,39,40,30,31,47)(25,38,34,32,29,26)(33,49,50,42,41,45)(35,37,48,44,43,46);(1,2,3)(5,6)(7,8)(10,11)(12,13)(14,15,18,24,31,41,46,47,44,34,20,19,25,33,43,23,29,36,45,50,28,38,27,37,42,32,17,21,26,35,40,48,49,39,30,22,16);(4,5,7,8,6)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 508);