C₃²×#93668 #72245355

order	4608
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃²×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₃²×#93668

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	#322
commutator	C₈	C₃²×C₄×C₁₆
Frattini, cosocle	#1311	C₂²×C₃²
socle, prime radical	C₂²×C₃²	#1313
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₃²×#1311 ◁ G

420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 4608

A minimal faithful permutation representation — 2 generators on 54 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,34,26,36,27,37)(19,28,29,38)(20,31)(25,40,41,48,42,47,39,35)(30,45,44,43)(32,46)(49,50,51);(17,18,19,25,20,23,30,35,21,27,28,41,32,37,43,40,22,26,29,42,31,36,44,48,33,24,38,39,46,34,45,47)(52,53,54)

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	C₃⋊D₃×#93668
maximal subgroup of	C₃⋊D₃×#93668, C₃²×#72224639
maximal normal subgroup of	C₃⋊D₃×#93668, C₃²×#72224639

GAP

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

gap> G := Group((1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23,24,34,26,36,27,37)(19,28,29,38)(20,31)(25,40,41,48,42,47,39,35)(30,45,44,43)(32,46)(49,50,51), (17,18,19,25,20,23,30,35,21,27,28,41,32,37,43,40,22,26,29,42,31,36,44,48,33,24,38,39,46,34,45,47)(52,53,54));

C32×#93668 #72245355