A₄×#12068 #72342871

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

A₄×#12068

Splittings

Fitting, Sylow

C₂²×#12068 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₄	A₄×#442
commutator	C₂²×C₈	C₂²×C₃×C₈
Frattini	C₄×C₈	C₂³×A₄
Fitting	C₂²×#12068	C₃
socle	C₂³	C₃×#1580
prime radical	A₄×#508	C₄
cosocle	C₂²×C₄×C₈	C₂³×C₃
hypercenter	#12068	A₄
hypocenter	C₂²	C₃×#12068
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #12068

1279 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4,3)(5,8,9,16,11,25,26,17,12,14,27,24,22,15,10,6)(7,13,28,33,20,31,35,29,21,32,34,36,19,30,18,23);(1,4)(2,3)(5,7)(6,13)(8,23)(9,18)(10,28)(11,19)(12,21)(14,29)(15,33)(16,30)(17,32)(20,22)(24,31)(25,36)(26,34)(27,35);(5,6,9,14,11,16,26,15,12,17,27,8,22,24,10,25)(7,13,18,29,19,30,34,33,21,32,35,23,20,31,28,36)

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×AGL(1,ℤ/32), #72342996, S₄×#12068
maximal normal subgroup of	A₄×AGL(1,ℤ/32), #72342996, S₄×#12068

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((2,4,3)(5,8,9,16,11,25,26,17,12,14,27,24,22,15,10,6)(7,13,28,33,20,31,35,29,21,32,34,36,19,30,18,23), (1,4)(2,3)(5,7)(6,13)(8,23)(9,18)(10,28)(11,19)(12,21)(14,29)(15,33)(16,30)(17,32)(20,22)(24,31)(25,36)(26,34)(27,35), (5,6,9,14,11,16,26,15,12,17,27,8,22,24,10,25)(7,13,18,29,19,30,34,33,21,32,35,23,20,31,28,36));

A4×#12068 #72342871