C₃×#8769 #10627376

order	768
nilpotency class	3
isoclinism	isoclinic class of #8760
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#8769

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#381
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 378 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)(5,8)(6,9)(10,11)(12,13,16,22,17,20,15,21)(14,23,24,27,25,18,26,19);(4,5,7,8)(6,9,10,11)(12,14)(13,18)(15,24)(16,26)(17,25)(19,21)(20,23)(22,27);(4,6)(5,9)(7,10)(8,11)(12,14)(13,19,20,27)(15,24)(16,26)(17,25)(18,21,23,22)

Conjugacy classes & character table

129 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1757);