C₃×#7366 #10625973

order	768
nilpotency class	5
isoclinism	isoclinic class of #1414
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₈ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7366

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#354
commutator	C₄²	C₂×C₃×C₈
Frattini, cosocle	#382	C₂²×C₃
prime radical	C₂³×C₃	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#382 ◁ G

158 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 50 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)(4,5,6,8,7,9,10,11)(12,13)(14,22,24,35)(15,26,27,42)(16,29,30,20)(17,21)(18,23,31,38)(19,28)(25,37,41,36)(32,39,33,40)(34,43);(12,14,16,24,17,25,30,41)(13,18,20,32,21,33,29,31)(15,23,28,39,27,40,43,38)(19,22,26,36,34,37,42,35)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 354);