C₃×#13592 #10632199

order	768
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃×#13592

Subgroups

	subgroup	G/H
center, socle	C₂×C₃	C₂×#353
commutator	C₄²	C₂²×C₃×C₄
Frattini, cosocle	C₄⋊Q₈	C₂³×C₃
prime radical	C₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₂×C₃×C₄ ◁ C₂²×C₃×C₄ ◁ C₃×#532 ◁ G

398 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 96

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

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

Conjugacy classes & character table

84 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 6580);