C₂×C₄×#6868 #14516205

order	1944
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄×C₉ · Schur multiplier (H₂) = C₂×C₃²

Aliases

direct product

C₂×C₄×#6868

Subgroups

	subgroup	G/H
center	C₂×C₃²×C₄	H(3,𝔽₃)
commutator	C₃²	C₂×C₃×C₄×C₉
Frattini, cosocle	C₂×C₃³	C₂²×C₃²
socle	C₂²×C₃²	C₂×H(3,𝔽₃)
prime radical	C₂²×C₃³	C₂×C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃³×C₄ ◁ G

288 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 2 generators on 42 points:

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

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, 599);