C₃²×#16861519 #49147100

order	36900
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃²×#16861519

Splittings

Fitting, socle, Hall	C₃²×C₅²×C₄₁ ⋊ C₄
derived, Hall	C₅²×C₄₁ ⋊ C₃²×C₄
Hall	C₃²×C₄₁ ⋊ #1043
Hall	C₃²×C₅² ⋊ #4223
Sylow	C₄₁ ⋊ C₃²×#1043
Sylow	C₅² ⋊ C₃²×#4223

Subgroups

	subgroup	G/H
center, hypercenter	C₃²	#16861519
commutator, hypocenter	C₅²×C₄₁	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃²×C₅²×C₄₁	C₄
prime radical	C₃²×C₅⋊D₂₀₅	C₂
cosocle	C₅⋊D₂₀₅	C₂×C₃²
LCS	G ▷ C₅²×C₄₁
UCS	C₁ ◁ C₃²

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 9225

A minimal faithful permutation representation — 3 generators on 57 points:

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

Conjugacy classes & character table

2340 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(36900, 501);

C32×#16861519 #49147100