C₃×#20269444 #34117482

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

Aliases

direct product

C₃×#20269444

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₃²×C₄
Hall	C₃×#13131975 ⋊ C₄
Hall	Dic₅₅₉ ⋊ C₃²
Sylow	C₄₃ ⋊ C₃×Dic₁₃⋊C₃
Sylow	C₁₃ ⋊ C₃×(C₄₃⋊C₃)⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#16809818
commutator, hypocenter	C₁₃×C₄₃	C₃²×C₄
Frattini	C₂	C₃×#16809818
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:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 85);

C3×#20269444 #34117482