C₃×#7412 #10626019

order	768
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7412

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#354
commutator	C₂×C₄²	C₂×C₃×C₄
Frattini, cosocle	#390	C₂²×C₃
prime radical	C₂²×C₃×D₄	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#390 ◁ G

208 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 46 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 — 2 generators on 27 points:

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 400);