C₃×#8476 #10627083

order	768
nilpotency class	3
isoclinism	isoclinic class of #8465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8476

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3512	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 678 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) = 384

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

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

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, 1464);