C₂×C₃⋊#16809810 #34117604

order	20124
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₂×C₃⋊#16809810
unambiguous semidirect product	(C₂×C₃)⋊#16809810

Splittings

Fitting, socle	C₂×C₃×C₁₃×C₄₃ ⋊ C₂×C₃
derived	C₃×C₁₃×C₄₃ ⋊ C₂²×C₃
Hall	C₃×#13131975 ⋊ C₂²
Hall	C₁₃×C₄₃ ⋊ C₂×C₃×D₃
Sylow	C₄₃ ⋊ C₂×D₃×C₁₃⋊C₃
Sylow	C₁₃ ⋊ C₂×C₃⋊(D₄₃⋊C₃)

Subgroups

	subgroup	G/H
center, hypercenter	C₂	C₃⋊#16809810
commutator, cosocle, hypocenter	C₃×C₁₃×C₄₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃×C₄₃	C₂×C₃
LCS	G ▷ C₃×C₁₃×C₄₃
UCS	C₁ ◁ C₂

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 3354

A minimal faithful permutation representation — 2 generators on 61 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)(45,46,48)(47,50,53)(49,52,51)(54,56,55)(58,59);(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)(44,45,47,46,49,53,50,54,56,48,51,55,52)(57,58,59)(60,61)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(20124, 207);

C2×C3⋊#16809810 #34117604