C₂×C₃×#19149851 #45686335

order	33300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃² · Schur multiplier (H₂) = C₂×C₃×C₅

Aliases

direct product

C₂×C₃×#19149851

Splittings

Fitting, socle	C₂×C₃×C₅²×C₃₇ ⋊ C₂×C₃
derived, Hall	C₅²×C₃₇ ⋊ C₂²×C₃²
Hall	C₃×C₅²⋊(C₃₇⋊C₃) ⋊ C₂²
Hall	C₂×C₅⋊D₁₈₅ ⋊ C₃²
Sylow	C₃₇ ⋊ C₂×C₃×(C₅⋊D₅)⋊C₃
Sylow	C₅² ⋊ C₂×C₃×D₃₇⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#19149851
commutator, cosocle, hypocenter	C₅²×C₃₇	C₂²×C₃²
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₅²×C₃₇	C₂×C₃
LCS	G ▷ C₅²×C₃₇
UCS	C₁ ◁ C₂×C₃

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 3330

A minimal faithful permutation representation — 2 generators on 57 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(33300, 608);

C2×C3×#19149851 #45686335