S₄×#13658 #72342918

order	6144
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

S₄×#13658

Splittings

Fitting

C₂²×#13658 ⋊ D₃

Subgroups

	subgroup	G/H
center	C₂	S₄×#2177
commutator	A₄×C₁₆	C₂³×C₄
Frattini	C₂×C₁₆	C₂³×S₄
Fitting	C₂²×#13658	D₃
socle	C₂³	D₃×#2177
prime radical	S₄×#2263	C₂
cosocle	C₂×A₄×C₁₆	C₂⁴
hypercenter	#13658	S₄
hypocenter	A₄	C₂×#13658
LCS	G ▷ A₄×C₁₆ ▷ C₈×A₄ ▷ C₄×A₄ ▷ C₂×A₄ ▷ A₄
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #13658

5304 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)(5,8,10,12)(6,14)(7,16)(9,25,27,20)(11,21,28,26)(15,32)(19,29)(22,35,34,30);(2,4)(5,7)(6,8)(9,24)(10,16)(11,15)(12,14)(13,30)(17,34)(18,22)(19,26)(20,23)(21,29)(25,36)(27,33)(28,32)(31,35);(5,6)(7,12)(8,16)(9,23)(10,14)(11,29)(13,22)(15,26)(17,30)(18,35)(19,28)(20,33)(21,32)(24,25)(27,36)(31,34);(6,13)(7,17)(8,12)(9,22)(11,26)(14,31)(15,33)(16,18)(19,36)(20,35)(21,28)(23,29)(24,32)(25,30)(27,34)

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

S4×#13658 #72342918