S₄×#12068 #72343116

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

S₄×#12068

Splittings

Fitting

C₂²×#12068 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₄	S₄×#442
commutator	C₈×A₄	C₂³×C₈
Frattini	C₄×C₈	C₂³×S₄
Fitting	C₂²×#12068	D₃
socle	C₂³	D₃×#1580
prime radical	S₄×#508	C₄
cosocle	C₄×C₈×A₄	C₂⁴
hypercenter	#12068	S₄
hypocenter	A₄	C₂×#12068
LCS	G ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #12068

5520 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Conjugacy classes & character table

230 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)(5,9,10,18,12,28,29,19,13,16,30,27,23,17,11,6)(7,14,24,26,8,15,31,32,22,34,36,35,21,33,20,25), (2,4)(5,8)(6,15)(7,12)(9,26)(10,24)(11,31)(13,21)(14,18)(16,35)(17,32)(19,33)(20,29)(22,23)(25,28)(27,34)(30,36), (5,7)(6,14)(8,23)(9,25)(10,20)(11,24)(12,21)(13,22)(15,27)(16,32)(17,26)(18,33)(19,34)(28,35)(29,36)(30,31), (5,6,10,16,12,18,29,17,13,19,30,9,23,27,11,28)(7,14,20,32,21,33,36,26,22,34,31,25,8,15,24,35));

S4×#12068 #72343116