C₃×#9862 #10628469

order	768
nilpotency class	3
isoclinism	isoclinic class of #1714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#9862

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#127
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
socle	C₂²×C₃	#390
prime radical	C₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

514 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 174 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)(12,13,15,18,17,21,16,20)(14,22,24,27,23,26,25,19);(4,5,9,11,7,10,6,8)(13,18,21,20)(14,23)(19,26,27,22)(24,25);(12,14,16,25,17,23,15,24)(13,19,18,22,21,27,20,26)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#141758, C₃×#213274
maximal normal subgroup of	C₃×#141758, C₃×#213274
maximal quotient of	C₃×#141837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 2850);