C₃×#20269442 #34117485

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

Aliases

direct product

C₃×#20269442

Splittings

derived, Hall	C₁₃×C₄₃ ⋊ C₃²×C₄
Hall	C₃×C₁₃⋊C₃×C₄₃ ⋊ C₄
Hall	Dic₅₅₉ ⋊ C₃²
Hall	C₃×C₁₃⋊C₃ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ C₃×Dic₁₃⋊C₃
Sylow	C₁₃ ⋊ C₃²×Dic₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃	(C₁₃⋊C₃)⋊D₄₃
commutator, hypocenter	C₁₃×C₄₃	C₃²×C₄
Frattini	C₂	C₃×(C₁₃⋊C₃)⋊D₄₃
Fitting, socle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
prime radical	C₂×C₃×C₁₃⋊C₃×C₄₃	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)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(45,49,50,46,47,51)(48,55,54,52,53,56)(57,60,59,58);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(44,45,48,47,51,54,53,55,56,50,49,52,46)(61,62,63)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 88);

C3×#20269442 #34117485