S₄×#7390 #72342985

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

Aliases

direct product

S₄×#7390

Splittings

Fitting

C₂²×#7390 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	S₄×#1333
commutator	A₄×C₁₆	C₂²×C₈
Frattini	#346	C₂²×S₄
Fitting	C₂²×#7390	D₃
socle	C₂³	D₃×#1333
prime radical	C₂×S₄×D₁₆	C₄
cosocle	A₄×#346	C₂³
hypercenter	#7390	S₄
hypocenter	A₄	C₂×#7390
LCS	G ▷ A₄×C₁₆ ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #7390

3111 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:

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

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

S4×#7390 #72342985