C₃×#7325 #10625932

order	768
nilpotency class	4
isoclinism	isoclinic class of #7304
automorphism	?
homology	Abelianization (H₁) = C₃×C₄² · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃×#7325

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	#342
commutator	C₂⁴	C₃×C₄²
Frattini, cosocle	C₂×#117	C₂²×C₃
prime radical	C₂⁴×C₃	C₄²
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#117 ◁ G

244 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 192

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 313);