C₃×#8495 #10627102

order	768
nilpotency class	3
isoclinism	isoclinic class of #8491
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#8495

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#379
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

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

normal subgroups only

This group has 620 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)(5,9,11,14)(10,18,19,17)(20,21)(22,30,33,26)(23,34,36,43)(24,29)(25,28)(27,35,41,37)(31,46)(32,45,50,40)(38,42)(39,47)(44,49)(48,51);(4,5)(6,10)(7,14)(8,11)(9,16)(12,17)(13,19)(15,18)(20,22,25,33)(21,26,28,30)(23,31,36,48)(24,32,38,50)(27,39,41,44)(29,40,42,45)(34,51,43,46)(35,47,37,49);(4,6,8,13)(5,10,11,19)(7,15,16,12)(9,17,14,18)(20,23)(21,27)(22,31)(24,37)(25,36)(26,39)(28,41)(29,43)(30,44)(32,49)(33,48)(34,42)(35,38)(40,46)(45,51)(47,50)

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 1483);