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C3×#16861518⋊C3 #49147000

order36900
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C3

Aliases

direct productC3×#16861518⋊C3
unambiguous semidirect product#16861518⋊C32
unambiguous semidirect product(C3×#16861518)⋊C3

Splittings

Fitting, socleC3×C52×C41 ⋊ C3×C4
derived, HallC52×C41 ⋊ C32×C4
HallC3×C41×C52⋊C3 ⋊ C4
Hall#16861518 ⋊ C32
HallC3×C52⋊C3 ⋊ #4223
SylowC41 ⋊ C3×#1043⋊C3
SylowC52 ⋊ C32×#4223

Subgroups

subgroupG/H
center, hypercenterC3#16861518⋊C3
commutator, hypocenterC52×C41C32×C4
FrattiniC1G
Fitting, socleC3×C52×C41C3×C4
prime radicalC3×C41⋊((C5⋊D5)⋊C3)C2
cosocleC5⋊D205C2×C32
LCSG ▷ C52×C41
UCSC1 ◁ C3

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 59
transitive degreeμt(G) = 1845

A minimal faithful permutation representation — 2 generators on 59 points:

(4,5,7)(6,17,18,9,10,14,11,8,13,12,15,16)(20,24,22,21)(23,33,27,26)(25,29,31,30)(28,46,36,35)(32,42,40,39)(34,38,44,43)(37,50,49,48)(41,58,53,52)(45,55,57,56)(47,51,54,59);(1,2,3)(4,6,12,9,11)(5,8,10,15,17)(7,13,14,16,18)(19,20,23,28,37,50,59,43,30,21,25,32,41,54,46,58,56,39,26,34,45,57,44,33,42,55,52,35,47,53,40,31,24,29,38,51,48,49,36,27,22)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36900, 401);