C₃×C₄×C₅²×C₄₃ #27166372

order	12900
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	?
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₅

Aliases

abelian

C₃×C₄×C₅²×C₄₃

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂	C₂×C₃×C₅²×C₄₃
socle, prime radical	C₂×C₃×C₅²×C₄₃	C₂
LCS	G ▷ C₁
UCS	C₁ ◁ G

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 — 2 generators on 60 points:

(1,2,3,4)(5,6,7)(8,9,10,11,12)(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);(13,14,15,16,17)

Conjugacy classes & character table

12900 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₅²×C₈×C₄₃, C₂×C₃×C₄×C₅²×C₄₃
maximal normal subgroup of	C₃×C₅²×C₈×C₄₃, C₂×C₃×C₄×C₅²×C₄₃
maximal quotient of	C₃×C₅²×C₈×C₄₃, C₂×C₃×C₄×C₅²×C₄₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 41);

C3×C4×C52×C43 #27166372