C₂×C₄×#6912 #14517797

order	1944
nilpotency class	3
isoclinism	isoclinic class of #6909
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂×C₃³

Aliases

direct product

C₂×C₄×#6912

Subgroups

	subgroup	G/H
center	C₂×C₃×C₄	C₃×H(3,𝔽₃)
commutator	C₃²	C₂×C₃³×C₄
Frattini, cosocle	C₂×C₃²	C₂²×C₃³
socle	C₂²×C₃	C₂×C₃×H(3,𝔽₃)
prime radical	C₂²×C₃³	C₂×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃³×C₄ ◁ G

496 conjugacy classes of subgroups; only the 264 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 264 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) = 33
transitive degree	μ_t(G) = 216

A minimal faithful permutation representation — 3 generators on 33 points:

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

Conjugacy classes & character table

280 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1944, 2191);