C₃×#26012816 #45686236

order	33300
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃×#26012816

Splittings

Fitting, socle	C₃×C₅²×C₃₇ ⋊ C₃×C₄
derived, Hall	C₅²×C₃₇ ⋊ C₃²×C₄
Hall	C₃×C₅²×C₃₇⋊C₃ ⋊ C₄
Hall	#16814877 ⋊ C₃²
Hall	C₃×C₃₇⋊C₃ ⋊ #1043
Sylow	C₃₇ ⋊ C₃²×#1043
Sylow	C₅² ⋊ C₃×#3866⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#26012816
commutator, hypocenter	C₅²×C₃₇	C₃²×C₄
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₃₇	C₃×C₄
prime radical	C₃×C₅⋊(C₅⋊(D₃₇⋊C₃))	C₂
cosocle	C₅⋊D₁₈₅	C₂×C₃²
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₃

288 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 50
transitive degree	μ_t(G) = 2775

A minimal faithful permutation representation — 3 generators on 50 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 509);

C3×#26012816 #45686236