C₄₃×C₃⋊#1039 #27166398

order	12900
isoclinism	isoclinic class of C₃⋊(C₅⋊D₅)
automorphism	?
homology	Abelianization (H₁) = C₄×C₄₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₄₃×C₃⋊#1039
unambiguous semidirect product	C₃⋊(C₄₃×#1039)

Splittings

derived, Hall	C₃×C₅² ⋊ C₄×C₄₃
Hall	C₃×C₅²×C₄₃ ⋊ C₄
Hall	C₅²×C₄₃ ⋊ Dic₃
Hall	C₃×C₄₃ ⋊ #1039
Sylow	C₅² ⋊ Dic₃×C₄₃
Sylow	C₃ ⋊ C₄₃×#1039

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₄₃	C₃⋊(C₅⋊D₅)
commutator, hypocenter	C₃×C₅²	C₄×C₄₃
Frattini	C₂	C₄₃×C₃⋊(C₅⋊D₅)
Fitting, socle, prime radical	C₂×C₃×C₅²×C₄₃	C₂
cosocle	C₂×C₃×C₅²	C₂×C₄₃
LCS	G ▷ C₃×C₅²
UCS	C₁ ◁ C₂×C₄₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 12900

A minimal faithful permutation representation — 3 generators on 60 points:

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

Conjugacy classes & character table

3354 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 67);

C43×C3⋊#1039 #27166398