A₄×Q₃₂ #73279

order	384
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

direct product	A₄×Q₃₂
unambiguous semidirect product	(C₂²×Q₃₂)⋊C₃

Splittings

Fitting, Sylow

C₂²×Q₃₂ ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	A₄×D₈
commutator, cosocle	C₂²×C₈	C₂²×C₃
Frattini	C₈	C₂²×A₄
Fitting	C₂²×Q₃₂	C₃
socle	C₂³	C₃×D₈
prime radical	C₂×A₄	D₈
hypercenter	Q₃₂	A₄
hypocenter	C₂²	C₃×Q₃₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 36 points:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×SD₆₄, A₄×Q₆₄, C₂×A₄×Q₃₂, …
maximal normal subgroup of	A₄×SD₆₄, A₄×Q₆₄, C₂×A₄×Q₃₂, …
maximal quotient of	A₄×#358, A₄×#366, #33930⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 5855);

A4×Q32 #73279

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

A₄×Q₃₂ #73279