C₃×#12251 #10630858

order	768
nilpotency class	4
isoclinism	isoclinic class of #12155
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#12251

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₃
prime radical	C₂³×C₃	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂×C₃×C₄⋊C₄ ◁ G

342 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 74 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)(4,5,7,10)(6,8,11,9)(12,13)(14,18)(15,25)(16,29)(17,21)(19,28)(20,30)(22,37)(23,36)(24,33)(26,41)(27,40)(31,38)(32,39)(34,42)(35,43);(5,8)(6,11)(9,10)(12,14,17,24)(13,18,21,33)(15,26,27,22)(16,23,30,39)(19,34,35,31)(20,36,29,32)(25,37,40,41)(28,42,43,38);(12,15)(13,19,21,35)(14,22)(16,28)(17,27)(18,31,33,34)(20,25,29,40)(23,38)(24,26)(30,43)(32,37,36,41)(39,42)

Conjugacy classes & character table

75 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5239);