C₃×#8540 #10627147

order	768
nilpotency class	3
isoclinism	isoclinic class of #8534
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#8540

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₂⁴	C₂²×C₃×C₄
Frattini, cosocle	C₂³×C₄	C₂³×C₃
prime radical	C₃×#534	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

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

normal subgroups only

This group has 718 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) = 51
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5)(6,12,14,9)(7,10)(8,11)(13,18,19,16)(15,17)(20,21,24,28,25,29,35,41)(22,26,32,38,31,37,45,49)(23,27,34,40,33,39,46,50)(30,42,44,51,43,36,47,48);(4,6)(5,9)(7,13)(8,14)(10,16)(11,12)(15,19)(17,18)(20,22)(21,26)(23,30)(24,32)(25,31)(27,36)(28,38)(29,37)(33,43)(34,44)(35,45)(39,42)(40,48)(41,49)(46,47)(50,51);(4,7,8,15)(5,10,11,17)(6,13,14,19)(9,16,12,18)(20,23)(21,27)(22,30)(24,34)(25,33)(26,36)(28,40)(29,39)(31,43)(32,44)(35,46)(37,42)(38,48)(41,50)(45,47)(49,51)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1528);