C₃×#7265 #10625872

order	768
nilpotency class	4
isoclinism	isoclinic class of #7262
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7265

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#340
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂³×C₈	C₂²×C₃
prime radical	C₂⁴×C₃	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

314 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 102 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 — 2 generators on 43 points:

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

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, 253);