C₂×C₄×#6899 #14517784

order	1944
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂×C₃

Aliases

direct product

C₂×C₄×#6899

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	C₃³
commutator	C₃²	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₃²	C₂²×C₃³
socle	C₂²×C₃²	C₂×C₃³
prime radical	C₂²×C₃³	C₂×C₃²
LCS	G ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ G

440 conjugacy classes of subgroups; only the 288 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 288 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) = 42
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2)(8,11,9)(12,13,14)(16,17,21)(18,25,37)(19,23,31)(20,24,32)(22,33,28)(26,36,40)(27,38,30)(29,35,41)(34,42,39);(3,4,5,6)(7,8,12,15,9,13,10,11,14)(16,18,26,20,27,39,29,28,19)(17,22,34,24,25,23,35,38,36)(21,30,31,32,33,40,41,37,42);(7,9,14,15,11,12,10,8,13)(16,19,28,29,39,27,20,26,18)(17,23,22,35,34,38,24,36,25)(21,31,33,41,42,30,32,40,37)

Conjugacy classes & character table

408 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2178);