C₃×#12162 #10630769

order	768
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#12162

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×C₄⋊C₈ ◁ G

354 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 82 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)(8,11)(12,13,17,21)(14,22,25,41)(15,26,30,40)(16,31,28,20)(18,33,35,23)(19,36,38,29)(24,42,32,37)(27,39,43,34);(4,5)(6,8)(7,9)(10,11)(12,14,17,25)(13,18,21,35)(15,27,30,43)(16,32,28,24)(19,37,38,42)(20,39,31,34)(22,40,41,26)(23,36,33,29);(4,6,7,10)(5,8,9,11)

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, 5150);