A₄×#13658 #72342783

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

A₄×#13658

Splittings

Fitting, Sylow

C₂²×#13658 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	A₄×#2177
commutator	C₂²×C₁₆	C₂²×C₃×C₄
Frattini	C₂×C₁₆	C₂³×A₄
Fitting	C₂²×#13658	C₃
socle	C₂³	C₃×#2177
prime radical	A₄×#2263	C₂
cosocle	C₂³×C₁₆	C₂³×C₃
hypercenter	#13658	A₄
hypocenter	C₂²	C₃×#13658
LCS	G ▷ C₂²×C₁₆ ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #13658

1229 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)(5,7,9,11)(6,13)(8,20,22,16)(10,17,23,21)(14,28)(15,25)(18,31,33,32)(24,36);(1,4)(2,3)(5,6)(7,15)(8,19)(9,13)(10,24)(11,25)(12,18)(14,21)(16,29)(17,28)(20,35)(22,34)(23,36)(26,32)(27,33)(30,31);(6,12)(7,11)(8,18)(10,21)(13,27)(14,29)(15,30)(16,31)(17,23)(19,24)(20,32)(22,33)(25,26)(28,35)(34,36)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#13658, A₄×AGL(1,ℤ/32), #72342975
maximal normal subgroup of	S₄×#13658, A₄×AGL(1,ℤ/32), #72342975

GAP

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

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

A4×#13658 #72342783