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C2×(C37⋊C3)⋊((C5⋊D5)⋊C3) #45686191

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

Aliases

direct productC2×(C37⋊C3)⋊((C5⋊D5)⋊C3)
unambiguous semidirect product(C2×C37⋊C3)⋊((C5⋊D5)⋊C3)
unambiguous semidirect product(C2×C5⋊(C5⋊(D37⋊C3)))⋊C3
unambiguous semidirect product(C2×#19149851)⋊C3
unambiguous semidirect product(C2×#19149853)⋊C3

Splittings

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

Subgroups

subgroupG/H
center, hypercenterC2(C37⋊C3)⋊((C5⋊D5)⋊C3)
commutator, cosocle, hypocenterC52×C37C22×C32
FrattiniC1G
Fitting, socleC2×C52×C37C2×C32
LCSG ▷ C52×C37
UCSC1 ◁ C2

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 54
transitive degreeμt(G) = 1110

A minimal faithful permutation representation — 2 generators on 54 points:

(3,4,6)(5,15,11,8,9,17)(7,12,16,13,14,10)(18,19)(20,22)(21,29)(23,25)(24,30)(26,28)(27,41)(31,33)(32,49)(34,35)(36,37)(38,39)(40,42)(43,45)(44,48)(46,47)(50,53)(51,52);(1,2)(3,5,10,16,8)(4,7,9,15,13)(6,11,12,14,17)(19,21,24)(20,23,25)(22,27,32)(26,31,40)(28,35,44)(29,33,38)(30,36,42)(34,43,51)(37,45,54)(39,47,52)(41,50,48)(46,53,49)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 464);