C₃×#13660 #10632267

order	768
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃×#13660

Subgroups

	subgroup	G/H
center	C₂×C₃×C₁₆	C₂³
commutator	C₂²	C₂²×C₃×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂³×C₃
socle, prime radical	C₂²×C₃	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₁₆ ◁ G

158 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 768

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

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

Conjugacy classes & character table

336 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6648);