C₃×#12702 #10631309

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×#12702

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	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂⁴×C₃ ◁ C₂³×C₃×Q₈ ◁ 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)(8,9,12,16,13,17,28,35)(10,18,21,36,19,32,26,38)(11,22,24,34,25,15,27,33)(14,29,31,23,30,37,39,20);(8,10)(9,14)(11,23)(12,26)(13,19)(15,32)(16,31)(17,30)(18,22)(20,25)(21,28)(24,29)(27,37)(33,38)(34,36)(35,39);(5,7)(10,19)(14,30)(18,32)(20,23)(21,26)(29,37)(31,39)(36,38)

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, 5690);