A₄×C₂₃×C₄₇ #27188447

order	12972
isoclinism	isoclinic class of A₄
automorphism	?
homology	Abelianization (H₁) = C₃×C₂₃×C₄₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×C₂₃×C₄₇
unambiguous semidirect product	C₂²⋊(C₃×C₂₃×C₄₇)
unambiguous semidirect product	(C₂²×C₂₃)⋊(C₃×C₄₇)
unambiguous semidirect product	(C₂²×C₂₃×C₄₇)⋊C₃

Splittings

Fitting, socle, Hall	C₂²×C₂₃×C₄₇ ⋊ C₃
derived, Sylow	C₂² ⋊ C₃×C₂₃×C₄₇
Hall	C₂²×C₄₇ ⋊ C₃×C₂₃
Hall	C₂²×C₂₃ ⋊ C₃×C₄₇

Subgroups

	subgroup	G/H
center, hypercenter	C₂₃×C₄₇	A₄
commutator, cosocle, hypocenter	C₂²	C₃×C₂₃×C₄₇
Frattini	C₁	G
Fitting, socle	C₂²×C₂₃×C₄₇	C₃
LCS	G ▷ C₂²
UCS	C₁ ◁ C₂₃×C₄₇

20 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 4324

A minimal faithful permutation representation — 2 generators on 74 points:

(2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51)(52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74);(1,4)(2,3)

Conjugacy classes & character table

4324 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(12972, 27);

A4×C23×C47 #27188447