C₃×#7091 #10625698

order	768
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#7091

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#327
commutator	C₂²×C₈	C₂×C₃×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂²×C₃
prime radical	C₂³×C₃	#102
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈ ◁ G

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

normal subgroups only

This group has 180 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) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 2 generators on 35 points:

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

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, 79);