C₃×#7645 #10626252

order	768
nilpotency class	3
isoclinism	isoclinic class of #7606
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃×#7645

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂×C₄²	C₂³×C₃
prime radical	C₂²×C₃×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄² ◁ G

630 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 384

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

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

Conjugacy classes & character table

156 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 633);