C₃×#7240 #10625847

order	768
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7240

Subgroups

	subgroup	G/H
center	C₃×C₄	H(3,ℤ/4)
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	#405	C₂²×C₃
socle	C₂×C₃	#1290
prime radical	C₂³×C₃	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₃×C₄ ◁ C₂×C₃×C₈ ◁ G

148 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 228);