C₃×#12704 #10631311

order	768
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃×#12704

Subgroups

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

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

normal subgroups only

This group has 962 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) = 39
transitive degree	μ_t(G) = 192

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3)(4,5)(6,7)(8,9,13,18,14,19,11,16)(10,20,23,38,24,35,21,37)(12,26,25,36,28,17,29,34)(15,30,31,27,32,39,33,22);(8,10,14,24)(9,15,19,32)(11,23,13,21)(12,27,28,22)(16,33,18,31)(17,35,26,20)(25,30,29,39)(34,37,36,38);(5,7)(8,11,14,13)(9,16,19,18)(10,21,24,23)(12,25,28,29)(15,31,32,33)(17,34,26,36)(20,37,35,38)(22,30,27,39)

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, 5692);