C₅²×Dic₃×C₄₃ #27166394

order	12900
isoclinism	isoclinic class of D₃
automorphism	?
homology	Abelianization (H₁) = C₄×C₅²×C₄₃ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₅²×Dic₃×C₄₃
unambiguous semidirect product	C₃⋊(C₄×C₅²×C₄₃)

Splittings

derived, Sylow	C₃ ⋊ C₄×C₅²×C₄₃
Hall	C₃×C₅²×C₄₃ ⋊ C₄
Hall	C₃×C₄₃ ⋊ C₄×C₅²
Hall	C₃×C₅² ⋊ C₄×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₅²×C₄₃	D₃
commutator, hypocenter	C₃	C₄×C₅²×C₄₃
Frattini	C₂	C₅²×D₃×C₄₃
Fitting, socle, prime radical	C₂×C₃×C₅²×C₄₃	C₂
cosocle	C₂×C₃	C₂×C₅²×C₄₃
LCS	G ▷ C₃
UCS	C₁ ◁ C₂×C₅²×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 — 2 generators on 60 points:

(2,3)(4,7,6,5)(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)(56,57,58,59,60);(1,2,3)(51,52,53,54,55)

Conjugacy classes & character table

6450 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 63);

C52×Dic3×C43 #27166394