C₃×#7260 #10625867

order	768
nilpotency class	4
isoclinism	isoclinic class of #1347
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃×#7260

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#102
commutator	C₂×C₈	C₃×C₄²
Frattini, cosocle	C₂³×C₈	C₂²×C₃
socle	C₂²×C₃	#340
prime radical	C₂⁴×C₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂³×C₃×C₄ ◁ C₂³×C₃×C₈ ◁ G

336 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 106 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)(12,13,16,21,18,22,17,32)(14,23,25,39,26,40,33,37)(15,27,29,38,31,42,30,20)(19,28,34,41,36,43,35,24);(5,8,10,9)(12,14)(13,19)(15,28)(16,25)(17,33)(18,26)(20,37)(21,34)(22,36)(23,27)(24,30)(29,41)(31,43)(32,35)(38,39)(40,42)

Conjugacy classes & character table

174 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 248);