C₃×#94506 #72245356

order	1536
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₃×C₈² · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×#94506

Subgroups

	subgroup	G/H
center	C₃×C₈	C₈²
commutator	C₈	C₃×C₈²
Frattini, cosocle	#1397	C₂²×C₃
socle	C₂×C₃	#7330
prime radical	C₂²×C₃	#1398
LCS	G ▷ C₈ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ G

118 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:

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

Conjugacy classes & character table

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

Relationships

Fitting subgroup of	D₃×#94506, #72245402, #72254989, …
maximal subgroup of	D₃×#94506, C₃×#72224644, #72245402, …
maximal normal subgroup of	D₃×#94506, C₃×#72224644, #72245402, …

GAP

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

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

C3×#94506 #72245356