C₃×#7155 #10625762

order	768
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#7155

Subgroups

	subgroup	G/H
center	C₃×C₄²	D₈
commutator	C₈	C₃×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
socle	C₂²×C₃	#340
prime radical	C₂³×C₃	#106
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₄² ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

222 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 2 generators on 43 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 143);