A₄×#7390 #72342853

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

Aliases

direct product

A₄×#7390

Splittings

Fitting, Sylow

C₂²×#7390 ⋊ C₃

Subgroups

	subgroup	G/H
center	C₂	A₄×#1333
commutator	C₂²×C₁₆	C₂×C₃×C₈
Frattini	#346	C₂²×A₄
Fitting	C₂²×#7390	C₃
socle	C₂³	C₃×#1333
prime radical	C₂×A₄×D₁₆	C₄
cosocle	C₂²×#346	C₂²×C₃
hypercenter	#7390	A₄
hypocenter	C₂²	C₃×#7390
LCS	G ▷ C₂²×C₁₆ ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂³ ▷ C₂²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #346 ◁ #7390

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

normal subgroups only

This group has 739 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)(6,13)(7,20)(8,23)(9,27)(10,29)(11,28)(14,18)(15,16)(17,24)(19,32)(21,25)(22,26)(30,35)(31,33)(34,36);(1,4)(2,3)(5,6,7,14,9,16,21,34,28,33,25,32,10,30,20,35,12,19,23,31,29,36,26,15,11,18,22,13,27,17,8,24)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×AGL(1,ℤ/32), S₄×#7390, #72343031
maximal normal subgroup of	A₄×AGL(1,ℤ/32), S₄×#7390, #72343031

GAP

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

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