C₃×#27162408 #49181769

order	37926
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃×#27162408

Splittings

Fitting, socle	C₃×C₇²×C₄₃ ⋊ C₂×C₃
derived, Hall	C₇²×C₄₃ ⋊ C₂×C₃²
Hall	C₃×C₄₃×#3861 ⋊ C₂
Hall	C₇⋊D₃₀₁ ⋊ C₃²
Hall	C₃×#3861 ⋊ D₄₃
Sylow	C₇² ⋊ C₃²×D₄₃
Sylow	C₄₃ ⋊ C₃×#64424

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#27162408
commutator, cosocle, hypocenter	C₇²×C₄₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃×C₇²×C₄₃	C₂×C₃
LCS	G ▷ C₇²×C₄₃
UCS	C₁ ◁ C₃

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 6321

A minimal faithful permutation representation — 3 generators on 60 points:

(5,8,9,6,7,10)(12,15,16,13,14,17)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)(45,46)(47,48)(49,50)(51,52)(53,54)(55,56)(57,58)(59,60);(1,2,3)(11,12,14,17,16,15,13)(18,19,21,23,25,27,29,31,33,35,37,39,41,43,45,47,49,51,53,55,57,59,60,58,56,54,52,50,48,46,44,42,40,38,36,34,32,30,28,26,24,22,20);(4,5,7,10,9,8,6)

Conjugacy classes & character table

1239 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(37926, 342);

C3×#27162408 #49181769