C₃×#12701 #10631308

order	768
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#12701

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	C₂×#99
commutator	C₂²×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×Q₈	C₂³×C₃
prime radical	C₂×C₃×#532	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×#41 ◁ G

1070 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 23
transitive degree	μ_t(G) = 96

A minimal faithful permutation representation — 3 generators on 23 points:

(1,2,3)(4,5)(6,7)(8,9,12,17,13,18,10,15)(11,16,19,23,20,14,21,22);(9,14)(11,20)(15,22)(16,18)(17,23)(19,21);(5,7)(8,10,13,12)(9,15,18,17)(11,19,20,21)(14,22,16,23)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 5689);