C₃×#12089 #10630696

order	768
nilpotency class	4
isoclinism	isoclinic class of #12087
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#12089

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#442
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₂×C₃×D₁₆	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#322 ◁ G

226 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 384

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

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#483961
maximal normal subgroup of	C₃×#483961

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5077);