C₃×#12698 #10631305

order	768
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃×#12698

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×D₄	C₂³×C₃
prime radical	C₂²×C₃×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

990 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 990 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 5686);