C₃×#8529 #10627136

order	768
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8529

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₃×#3553	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 648 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)(5,10)(6,13)(7,8)(12,15)(14,16)(17,18)(20,21)(22,30,33,26)(23,34,35,45)(24,29)(25,28)(27,36,42,38)(31,47)(32,44,37,41)(39,43)(40,49)(46,50)(48,51);(4,5,9,12)(6,11,14,19)(7,15,8,10)(13,18,16,17)(20,22,25,33)(21,26,28,30)(23,31,35,48)(24,37,39,32)(27,40,42,46)(29,44,43,41)(34,47,45,51)(36,50,38,49);(4,6)(5,11)(7,13)(8,16)(9,14)(10,17)(12,19)(15,18)(20,23)(21,27)(22,31)(24,38)(25,35)(26,40)(28,42)(29,45)(30,46)(32,50)(33,48)(34,43)(36,39)(37,49)(41,47)(44,51)

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, 1517);