C₃×#10015 #10628622

order	768
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#10015

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#392
prime radical	C₃×#2397	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

696 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 696 conjugacy classes of subgroups — too many to draw. Only its 122 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 — 3 generators on 27 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 3003);