C₃×#7144 #10625751

order	768
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃×#7144

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	#95
commutator	C₂×C₄	C₃×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₃
socle	C₂²×C₃	#336
prime radical	C₂³×C₃	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄×C₈ ◁ G

210 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 98 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)(6,12,14,18)(7,10)(8,11)(9,13,16,19)(15,17)(20,21,23,27,24,29,34,41,25,28,35,40,36,42,47,50)(22,30,32,43,33,44,46,51,31,26,37,38,45,39,48,49);(4,6)(5,9,11,16)(7,13)(8,14)(10,12,17,18)(15,19)(20,22)(21,26,28,30)(23,32)(24,33)(25,31)(27,38,40,43)(29,39,42,44)(34,46)(35,37)(36,45)(41,49,50,51)(47,48)

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 132);