C₃×#12259 #10630866

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

Aliases

direct product

C₃×#12259

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

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

normal subgroups only

This group has 332 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) = 35
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 35 points:

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

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, 5247);