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C2×C3×C5⋊(C5⋊(D37⋊C3)) #45686341

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

Aliases

direct productC2×C3×C5⋊(C5⋊(D37⋊C3))
unambiguous semidirect product(C2×C5)⋊(C3×C5⋊(D37⋊C3))
unambiguous semidirect product(C2×C37⋊C3)⋊(C3×C5⋊D5)

Splittings

Fitting, socleC2×C3×C52×C37 ⋊ C2×C3
derived, HallC52×C37 ⋊ C22×C32
HallC3×C52×C37⋊C3 ⋊ C22
HallC2×C5⋊D185 ⋊ C32
HallC3×C37⋊C3 ⋊ C2×C5⋊D5
SylowC37 ⋊ C2×C32×C5⋊D5
SylowC52 ⋊ C2×C3×D37⋊C3

Subgroups

subgroupG/H
center, hypercenterC2×C3C5⋊(C5⋊(D37⋊C3))
commutator, cosocle, hypocenterC52×C37C22×C32
FrattiniC1G
Fitting, socleC2×C3×C52×C37C2×C3
LCSG ▷ C52×C37
UCSC1 ◁ C2×C3

480 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 52
transitive degreeμt(G) = 5550

A minimal faithful permutation representation — 3 generators on 52 points:

(7,8)(9,10)(12,13)(14,15)(17,21,22,18,19,23)(20,29,30,24,25,38)(26,41,42,32,33,49)(27,40,36,34,31,28)(35,51,52,44,43,47)(37,39,50,46,45,48);(1,2)(3,4,5)(11,12,14,15,13)(16,17,20,26,33,43,48,49,46,36,22,21,27,35,45,25,31,38,47,52,30,40,29,39,44,34,19,23,28,37,42,50,51,41,32,24,18);(6,7,9,10,8)

Conjugacy classes & character table

1152 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 614);