C₃×#7409 #10626016

order	768
nilpotency class	5
isoclinism	isoclinic class of #7408
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7409

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	AΣL(1,𝔽₁₆)
commutator	C₂×C₄⋊C₄	C₂×C₃×C₄
Frattini, cosocle	#391	C₂²×C₃
prime radical	C₃×#537	C₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#391 ◁ G

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

normal subgroups only

This group has 268 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) = 96

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 397);