C₃²×#7390 #72255453

order	2304
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₈ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#7390

Subgroups

	subgroup	G/H
center, socle	C₂×C₃²	#1333
commutator	C₁₆	C₂×C₃²×C₈
Frattini, cosocle	#346	C₂²×C₃²
prime radical	C₂×C₃²×D₁₆	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃² ◁ C₂×C₃²×C₄ ◁ C₃²×C₄×C₈ ◁ C₃²×#346 ◁ G

384 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 38
transitive degree	μ_t(G) = 288

A minimal faithful permutation representation — 2 generators on 38 points:

(1,2,3,9,5,11,4,18,20,31,19,30,6,25,15,10,8,14,17,28,24,29,23,32,7,13,16,27,21,12,22,26)(33,34,35);(2,9)(3,15)(4,19)(5,21)(6,24)(7,20)(10,29)(11,26)(12,18)(13,30)(14,28)(16,23)(17,22)(25,32)(27,31)(36,37,38)

Conjugacy classes & character table

306 conjugacy classes. The character table is too large to display.

Relationships

Fitting subgroup of	C₃⋊D₃×#7390, #72255470
maximal subgroup of	C₃⋊D₃×#7390, C₃²×AGL(1,ℤ/32), #72255470
maximal normal subgroup of	C₃⋊D₃×#7390, C₃²×AGL(1,ℤ/32), #72255470
maximal quotient of	C₃²×#93757

GAP

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

gap> G := Group((1,2,3,9,5,11,4,18,20,31,19,30,6,25,15,10,8,14,17,28,24,29,23,32,7,13,16,27,21,12,22,26)(33,34,35), (2,9)(3,15)(4,19)(5,21)(6,24)(7,20)(10,29)(11,26)(12,18)(13,30)(14,28)(16,23)(17,22)(25,32)(27,31)(36,37,38));