S₄×#7389 #72342928

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

S₄×#7389

Splittings

Fitting

C₂²×#7389 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	S₄×#1333
commutator	A₄×C₁₆	C₂²×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#7389	D₃
socle	C₂³	D₃×#1333
prime radical	S₄×#508	C₄
cosocle	A₄×#346	C₂³
hypercenter	#7389	S₄
hypocenter	A₄	C₂×#7389
LCS	G ▷ A₄×C₁₆ ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #7389

2736 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:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×AGL(1,ℤ/32)
maximal normal subgroup of	S₄×AGL(1,ℤ/32)

GAP

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

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

S4×#7389 #72342928