C₃×#10632 #10629239

order	768
nilpotency class	3
isoclinism	isoclinic class of #10631
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#10632

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#397
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂³×C₃	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ G

310 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 192

A minimal faithful permutation representation — 3 generators on 43 points:

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

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, 3620);