C₂×C₃×#1843 #10639397

order	768
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×C₃×#1843

Subgroups

	subgroup	G/H
center	C₂³×C₃×C₄	C₂³
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle, prime radical	C₂⁴×C₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₄ ◁ G

1230 conjugacy classes of subgroups; only the 702 normal subgroups are shown.

normal subgroups only

This group has 1230 conjugacy classes of subgroups — too many to draw. Only its 702 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) = 29
transitive degree	μ_t(G) = 768

A minimal faithful permutation representation — 4 generators on 29 points:

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

Conjugacy classes & character table

264 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(768, 13778);