C₃×#13056 #10631663

order	768
nilpotency class	4
isoclinism	isoclinic class of #2202
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃×#13056

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₃×C₄
Frattini, cosocle	C₂²×Q₈	C₂³×C₃
prime radical	C₂×C₃×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃ ◁ C₂²×C₃×Q₈ ◁ G

632 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 632 conjugacy classes of subgroups — too many to draw. Only its 118 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)(5,7)(8,9,11,19,14,20,12,17)(10,21,24,15,25,32,22,33)(13,18,27,16,29,34,26,35)(23,36,37,39,28,30,38,31);(4,5,6,7)(8,10)(9,15)(11,22)(12,24)(13,28)(14,25)(16,30)(17,32)(18,31)(19,21)(20,33)(23,29)(26,38)(27,37)(34,39)(35,36);(8,11,14,12)(9,16,20,35)(10,22,25,24)(13,27,29,26)(15,30,33,36)(17,34,19,18)(21,31,32,39)(23,38,28,37)

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, 6044);