C₃²×#7463 #72254864

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

Aliases

direct product

C₃²×#7463

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#367
commutator	C₁₆	C₂×C₃²×C₈
Frattini, cosocle	C₄×C₁₆	C₂²×C₃²
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ C₃²×C₄×C₈ ◁ C₃²×C₄×C₁₆ ◁ G

252 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 46
transitive degree	μ_t(G) = 2304

A minimal faithful permutation representation — 2 generators on 46 points:

(1,2,3,5,4,6,7,8)(10,15,17,28)(11,20)(12,24)(13,25)(16,31)(18,32,35,40)(19,38,36,30)(21,26)(22,34)(23,33)(27,39,37,29)(41,42,43);(9,10,16,32,12,18,11,15,25,36,20,29,21,27,31,38,14,17,34,40,26,35,23,28,13,19,33,39,24,37,22,30)(44,45,46)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#7463, #72269686
maximal subgroup of	C₃²×#151085, C₃⋊D₃×#7463, #72269686
maximal normal subgroup of	C₃²×#151085, C₃⋊D₃×#7463, #72269686

GAP

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

gap> G := Group((1,2,3,5,4,6,7,8)(10,15,17,28)(11,20)(12,24)(13,25)(16,31)(18,32,35,40)(19,38,36,30)(21,26)(22,34)(23,33)(27,39,37,29)(41,42,43), (9,10,16,32,12,18,11,15,25,36,20,29,21,27,31,38,14,17,34,40,26,35,23,28,13,19,33,39,24,37,22,30)(44,45,46));

C32×#7463 #72254864