C₃×#26012817 #45686237

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

Aliases

direct product

C₃×#26012817

Splittings

Fitting, socle	C₃×C₅²×C₃₇ ⋊ C₃×C₄
derived, Hall	C₅²×C₃₇ ⋊ C₃²×C₄
Hall	C₃×C₅²×C₃₇⋊C₃ ⋊ C₄
Hall	#16814878 ⋊ C₃²
Hall	C₃×C₃₇⋊C₃ ⋊ #1043
Sylow	C₃₇ ⋊ C₃²×#1043
Sylow	C₅² ⋊ C₃×#3866⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#26012817
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)(15,31,25,16,17,37,18,20,23,21,24,30)(19,50,47,26,27,38,28,39,42,40,43,32)(22,44,45,34,35,46,36,49,48,33,29,41);(1,2,3)(9,10,11,13,12)(14,15,19,29,45,34,16,22,20,32,23,24,26,41,37,27,39,47,44,49,42,50,43,30,46,40,17,25,38,31,36,21,33,48,35,28,18);(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, 510);

C3×#26012817 #45686237