C₃×#94754 #72254995

order	1536
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#94754

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	#7463
commutator	C₃₂	C₂×C₃×C₈
Frattini, cosocle	#1397	C₂²×C₃
prime radical	C₂²×C₃	#1426
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×C₄×C₁₆ ◁ C₃×#1397 ◁ G

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 67
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 67 points:

(1,2,3,9,15,32,49,8,5,13,17,34,4,11,14,29,24,37,53,27,19,31,46,26,20,40,45,39,48,59,55,56,7,12,16,33,51,60,64,28,25,42,52,61,22,41,44,57,6,10,18,35,50,58,63,30,23,38,54,62,21,36,47,43)(65,66,67);(2,8,10,26,12,28,37,30)(3,14,16,44)(4,19)(5,23)(6,24)(9,31,27,32,33,58,35,60)(11,39,36,61,41,62,59,34)(13,43,38,29,42,56,40,57)(15,48)(17,49,52,64)(18,55,53,47)(20,25)(21,51)(22,50)(45,63,54,46)

Conjugacy classes & character table

168 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	D₃×#94754, #72255030
maximal subgroup of	C₃×#72224652, D₃×#94754, #72255030
maximal normal subgroup of	C₃×#72224652, D₃×#94754, #72255030

GAP

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

gap> G := Group((1,2,3,9,15,32,49,8,5,13,17,34,4,11,14,29,24,37,53,27,19,31,46,26,20,40,45,39,48,59,55,56,7,12,16,33,51,60,64,28,25,42,52,61,22,41,44,57,6,10,18,35,50,58,63,30,23,38,54,62,21,36,47,43)(65,66,67), (2,8,10,26,12,28,37,30)(3,14,16,44)(4,19)(5,23)(6,24)(9,31,27,32,33,58,35,60)(11,39,36,61,41,62,59,34)(13,43,38,29,42,56,40,57)(15,48)(17,49,52,64)(18,55,53,47)(20,25)(21,51)(22,50)(45,63,54,46));

C3×#94754 #72254995