C₃×#20269435 #34117486

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

Aliases

direct product

C₃×#20269435

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₃²×C₄
Hall	C₃×#13131976 ⋊ C₄
Hall	C₁₃×Dic₄₃ ⋊ C₃²
Hall	Dic₄₃ ⋊ C₃×C₁₃⋊C₃
Sylow	C₄₃ ⋊ C₃×C₄×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₃×(C₄₃⋊C₃)⋊C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	#16809809
commutator, hypocenter	C₁₃×C₄₃	C₃²×C₄
Frattini	C₂	C₃×#16809809
Fitting, socle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
prime radical	C₂×C₃×#13131976	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)(11,27,28,17,18,36)(12,26,22,20,16,13)(19,32,34,23,25,37)(21,38,39,30,29,33)(31,41,42,40,35,43)(45,46,48)(47,50,53)(49,52,51)(54,56,55)(57,60,59,58);(1,2,5,11,19,22,7,6,12,21,31,39,15,26,14,25,35,42,28,37,38,27,18,29,34,36,43,41,32,10,16,24,33,40,30,20,4,8,13,23,17,9,3)(44,45,47,46,49,53,50,54,56,48,51,55,52)(61,62,63)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 89);

C3×#20269435 #34117486