C₃×#12412 #10631019

order	768
nilpotency class	4
isoclinism	isoclinic class of #12324
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12412

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#447
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
prime radical	C₂×C₃×D₈	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#446 ◁ G

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

normal subgroups only

This group has 280 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) = 51
transitive degree	μ_t(G) = 384

A minimal faithful permutation representation — 3 generators on 51 points:

(1,2,3)(4,5,10,14)(6,15,7,18)(8,13,9,12)(11,17,19,16)(20,21,26,30)(22,31,34,49)(23,28,25,40)(24,36,39,47)(27,42,44,33)(29,35,48,38)(32,50,41,46)(37,43,51,45);(4,6,10,7)(5,11,14,19)(8,17,9,16)(12,18,13,15)(20,22,26,34)(21,27,30,44)(23,32,25,41)(24,37,39,51)(28,43,40,45)(29,46,48,50)(31,47,49,36)(33,35,42,38);(4,7,10,6)(5,11,14,19)(8,16,9,17)(12,18,13,15)(20,23,26,25)(21,28,30,40)(22,32,34,41)(24,38,39,35)(27,43,44,45)(29,47,48,36)(31,46,49,50)(33,51,42,37)

Conjugacy classes & character table

93 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5400);