A₄×#7389 #72342833

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

A₄×#7389

Splittings

Fitting, Sylow

C₂²×#7389 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	A₄×#1333
commutator	C₂²×C₁₆	C₂×C₃×C₈
Frattini	#346	C₂²×A₄
Fitting	C₂²×#7389	C₃
socle	C₂³	C₃×#1333
prime radical	A₄×#508	C₄
cosocle	C₂²×#346	C₂²×C₃
hypercenter	#7389	A₄
hypocenter	C₂²	C₃×#7389
LCS	G ▷ C₂²×C₁₆ ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #7389

657 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	S₄×#7389, A₄×AGL(1,ℤ/32), #72342952
maximal normal subgroup of	S₄×#7389, A₄×AGL(1,ℤ/32), #72342952

GAP

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

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