C₃×#13280 #10631887

order	768
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#13280

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#102
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₈	C₂³×C₃
prime radical	C₂³×C₃	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×C₄×C₈ ◁ G

334 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 334 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 384

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

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

Conjugacy classes & character table

138 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6268);