C₃×#14170 #10632777

order	768
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

direct product

C₃×#14170

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₃	C₂⁴
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂⁴	C₂⁴×C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃ ◁ G

1494 conjugacy classes of subgroups; only the 494 normal subgroups are shown.

normal subgroups only

This group has 1494 conjugacy classes of subgroups — too many to draw. Only its 494 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) = 768

A minimal faithful permutation representation — 4 generators on 27 points:

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

Conjugacy classes & character table

228 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7158);