C₃×#14663 #10633270

order	768
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

direct product

C₃×#14663

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

1462 conjugacy classes of subgroups; only the 462 normal subgroups are shown.

normal subgroups only

This group has 1462 conjugacy classes of subgroups — too many to draw. Only its 462 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) = 35
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 35 points:

(1,2,3)(12,13,15,17)(14,16,18,19)(21,25)(26,33)(27,30)(34,35);(5,7)(8,9,10,11)(14,18)(16,19)(20,21,24,25)(22,26,29,33)(23,30,32,27)(28,35,31,34);(4,5)(6,7)(8,9)(10,11)(12,13)(14,16)(15,17)(18,19)(20,22,24,29)(21,26,25,33)(23,31,32,28)(27,35,30,34);(12,14)(13,16)(15,18)(17,19)(20,23,24,32)(21,27,25,30)(22,28,29,31)(26,34,33,35)

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 7651);