C₃×#7066 #10625673

order	768
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7066

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#99
commutator	C₂²×C₄	C₂×C₃×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₃
socle	C₂²×C₃	#324
prime radical	C₂³×C₃	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈ ◁ G

176 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 43
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 54);