C₃×#8705 #10627312

order	768
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#8705

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₂⁴×C₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

486 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 43
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 43 points:

(1,2,3)(4,5)(6,9)(7,8)(10,11)(12,13)(14,17)(15,22,23,18)(16,19)(20,26,27,24)(21,25)(28,29,31,35,32,34,38,41)(30,33,37,40,36,39,42,43);(5,8)(9,11)(12,14,16,21)(13,17,19,25)(15,20,23,27)(18,24,22,26)(28,30)(29,33)(31,37)(32,36)(34,39)(35,40)(38,42)(41,43);(4,6,7,10)(5,9,8,11)(12,15)(13,18)(14,20)(16,23)(17,24)(19,22)(21,27)(25,26)(29,34)(30,36)(31,38)(40,43)

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, 1693);