C₃×#19078 #10637685

order	768
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#19078

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

944 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 27 points:

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

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, 12066);