C₃×#94509 #72245377

order	1536
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₁₆ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#94509

Subgroups

	subgroup	G/H
center	C₃×C₈	#322
commutator	C₈	C₃×C₄×C₁₆
Frattini, cosocle	#1397	C₂²×C₃
socle	C₂×C₃	#7332
prime radical	C₂²×C₃	#1398
LCS	G ▷ C₈ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₃×#1397 ◁ G

114 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:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₃×#94509, #72245409, #72269614, …
maximal subgroup of	D₃×#94509, C₃×#72224644, #72245409, …
maximal normal subgroup of	D₃×#94509, C₃×#72224644, #72245409, …

GAP

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

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

C3×#94509 #72245377