C₃×#20269440 #34117484

order	20124
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄ · Schur multiplier (H₂) = C₃

Aliases

direct product

C₃×#20269440

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₃²×C₄
Hall	C₃×#13131975 ⋊ C₄
Hall	C₄₃×Dic₁₃ ⋊ C₃²
Hall	Dic₁₃ ⋊ C₃×C₄₃⋊C₃
Sylow	C₄₃ ⋊ C₃×Dic₁₃⋊C₃
Sylow	C₁₃ ⋊ C₃×C₄×C₄₃⋊C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#16809814
commutator, hypocenter	C₁₃×C₄₃	C₃²×C₄
Frattini	C₂	C₃×#16809814
Fitting, socle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
prime radical	C₂×C₃×#13131975	C₂
cosocle	C₂×C₁₃×C₄₃	C₂×C₃²
LCS	G ▷ C₁₃×C₄₃
UCS	C₁ ◁ C₂×C₃

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 63
transitive degree	μ_t(G) = 6708

A minimal faithful permutation representation — 2 generators on 63 points:

(2,3,5)(4,7,12)(6,10,9)(8,14,22)(11,17,19)(13,20,16)(15,24,30)(18,26,31)(21,29,28)(23,32,25)(27,36,38)(33,39,37)(34,40,35)(41,43,42)(45,49,50,46,47,51)(48,55,54,52,53,56)(57,60,59,58);(1,2,4,8,15,17,3,6,11,18,27,26,7,13,21,30,38,36,14,23,33,40,22,31,24,34,41,43,32,12,19,28,37,42,39,20,5,9,16,25,35,29,10)(44,45,48,47,51,54,53,55,56,50,49,52,46)(61,62,63)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 87);

C3×#20269440 #34117484