C₃×#24926546 #45659484

order	32850
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃×C₅

Aliases

direct product

C₃×#24926546

Splittings

Fitting, socle	C₃×C₅²×C₇₃ ⋊ C₂×C₃
derived, Hall	C₅²×C₇₃ ⋊ C₂×C₃²
Hall	C₃×C₅²⋊(C₇₃⋊C₃) ⋊ C₂
Hall	C₅⋊D₃₆₅ ⋊ C₃²
Sylow	C₇₃ ⋊ C₃×(C₅⋊D₅)⋊C₃
Sylow	C₅² ⋊ C₃×D₇₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#24926546
commutator, cosocle, hypocenter	C₅²×C₇₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₇₃	C₂×C₃
LCS	G ▷ C₅²×C₇₃
UCS	C₁ ◁ C₃

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 91
transitive degree	μ_t(G) = 3285

A minimal faithful permutation representation — 2 generators on 91 points:

(4,5,7)(6,16,12,9,10,18)(8,13,17,14,15,11)(20,24,25,21,22,26)(23,32,33,27,28,42)(29,45,46,35,36,59)(30,44,40,38,34,31)(37,64,65,48,49,78)(39,62,63,51,47,56)(41,43,61,53,54,57)(50,66,75,52,67,81)(55,60,80,68,69,76)(58,85,83,72,73,71)(70,79,86,74,82,90)(77,91,89,87,88,84);(1,2,3)(4,6,11,17,9)(5,8,10,16,14)(7,12,13,15,18)(19,20,23,29,37,50,51,38,22,26,31,41,55,70,72,54,28,34,42,56,71,84,87,69,36,47,57,59,75,86,91,82,49,66,73,76,78,65,80,85,67,64,79,88,90,81,46,61,62,45,60,77,89,83,63,33,44,32,43,58,74,68,53,40,25,24,30,39,52,48,35,27,21)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32850, 87);

C3×#24926546 #45659484