A₄×C₂₅×C₄₃ #27166363

order	12900
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₅×C₄₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₂₅×C₄₃
unambiguous semidirect product	C₂²⋊(C₃×C₂₅×C₄₃)
unambiguous semidirect product	(C₂²×C₂₅)⋊(C₃×C₄₃)

Splittings

Fitting, Hall	C₂²×C₂₅×C₄₃ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₂₅×C₄₃
Hall	C₂²×C₄₃ ⋊ C₃×C₂₅
Hall	C₂²×C₂₅ ⋊ C₃×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂₅×C₄₃	A₄
commutator, hypocenter	C₂²	C₃×C₂₅×C₄₃
Frattini	C₅	C₅×A₄×C₄₃
Fitting	C₂²×C₂₅×C₄₃	C₃
socle	C₂²×C₅×C₄₃	C₃×C₅
prime radical	C₅×A₄×C₄₃	C₅
cosocle	C₂²×C₅	C₃×C₅×C₄₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂₅×C₄₃

30 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 4300

A minimal faithful permutation representation — 2 generators on 72 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47)(48,49,51,54,58,50,52,55,59,63,53,56,60,64,67,57,61,65,68,70,62,66,69,71,72);(1,4)(2,3)

Conjugacy classes & character table

4300 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12900, 32);

A4×C25×C43 #27166363