C₂×C₄×#6896 #14517781

order	1944
nilpotency class	2
isoclinism	isoclinic class of C₃²⋊H(3,𝔽₃)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃³×C₄ · Schur multiplier (H₂) = C₂×C₉

Aliases

direct product

C₂×C₄×#6896

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

480 conjugacy classes of subgroups; only the 312 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 312 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) = 24
transitive degree	μ_t(G) = 648

A minimal faithful permutation representation — 3 generators on 24 points:

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

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, 2175);