C₅×D₃×D₅×C₄₃ #27166456

order	12900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅×C₄₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	C₅×D₃×D₅×C₄₃
unambiguous semidirect product	(C₅×D₃)⋊(C₂×C₅×C₄₃)
unambiguous semidirect product	(C₃×D₅)⋊(C₂×C₅×C₄₃)
unambiguous semidirect product	D₁₅⋊(C₂×C₅×C₄₃)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₄₃ ⋊ C₂²
derived	C₃×C₅ ⋊ C₂²×C₅×C₄₃
Hall	C₅²×C₄₃ ⋊ C₂×D₃
Hall	C₃×C₄₃ ⋊ C₂×C₅×D₅
Hall	C₃×C₅² ⋊ C₂²×C₄₃
Sylow	C₅² ⋊ C₂×D₃×C₄₃
Sylow	C₃ ⋊ C₂×C₅×D₅×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅×C₄₃	D₃×D₅
commutator, cosocle, hypocenter	C₃×C₅	C₂²×C₅×C₄₃
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₄₃	C₂²
LCS	G ▷ C₃×C₅
UCS	C₁ ◁ C₅×C₄₃

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 1290

A minimal faithful permutation representation — 2 generators on 56 points:

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

Conjugacy classes & character table

2580 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 125);

C5×D3×D5×C43 #27166456