C₅²×D₃×D₄₃ #27166452

order	12900
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅² · Schur multiplier (H₂) = C₂×C₅

Aliases

direct product	C₅²×D₃×D₄₃
unambiguous semidirect product	(D₃×C₄₃)⋊(C₂×C₅²)
unambiguous semidirect product	(C₃×D₄₃)⋊(C₂×C₅²)
unambiguous semidirect product	D₁₂₉⋊(C₂×C₅²)

Splittings

Fitting, socle, Hall	C₃×C₅²×C₄₃ ⋊ C₂²
derived, Hall	C₃×C₄₃ ⋊ C₂²×C₅²
Hall	C₅²×C₄₃ ⋊ C₂×D₃
Hall	C₃×C₅² ⋊ C₂×D₄₃
Sylow	C₄₃ ⋊ C₂×C₅²×D₃
Sylow	C₃ ⋊ C₂×C₅²×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅²	D₃×D₄₃
commutator, cosocle, hypocenter	C₃×C₄₃	C₂²×C₅²
Frattini	C₁	G
Fitting, socle	C₃×C₅²×C₄₃	C₂²
LCS	G ▷ C₃×C₄₃
UCS	C₁ ◁ C₅²

160 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 3225

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

Conjugacy classes & character table

1725 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 121);

C52×D3×D43 #27166452