C₃×#9505 #10628112

order	768
nilpotency class	3
isoclinism	isoclinic class of #9495
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#9505

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#386
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂²×C₃×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

684 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 27
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 27 points:

(1,2,3)(4,5)(6,10,11,8)(7,9)(12,13,15,19,16,18,20,26)(14,17,22,24,21,25,23,27);(4,5)(6,8)(7,9)(10,11)(12,14,20,22,16,21,15,23)(13,17,19,27,18,25,26,24);(4,6)(5,8)(7,11)(9,10)

Conjugacy classes & character table

111 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2493);