C₃×#12519 #10631126

order	768
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂×C₈

Aliases

direct product

C₃×#12519

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

360 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 110 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)(5,8)(6,9)(10,11)(13,17)(14,18)(15,21)(19,23)(20,27)(22,26)(24,25);(12,13,18,25,15,20,22,23,16,19,26,27,21,24,14,17);(4,5,9,11,7,10,6,8)(13,19)(17,23)(20,24)(25,27)

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#491469, D₃×#12519, #72225923, …
maximal normal subgroup of	C₃×#491469, D₃×#12519, #72225923, …
maximal quotient of	C₃×#123412, C₃×#123419

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5507);