C₂×C₃×#6868 #13090173

order	1458
nilpotency class	3
isoclinism	isoclinic class of AΣL(1,𝔽₂₇)
automorphism	?
homology	Abelianization (H₁) = C₂×C₃²×C₉ · Schur multiplier (H₂) = C₃⁴

Aliases

direct product

C₂×C₃×#6868

Subgroups

	subgroup	G/H
center, socle	C₂×C₃³	H(3,𝔽₃)
commutator	C₃²	C₂×C₃²×C₉
Frattini, cosocle	C₃³	C₂×C₃³
prime radical	C₂×C₃⁴	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃³ ◁ C₂×C₃⁴ ◁ G

476 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 476 conjugacy classes of subgroups — too many to draw. Only its 188 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) = 41
transitive degree	μ_t(G) = 486

A minimal faithful permutation representation — 3 generators on 41 points:

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

Conjugacy classes & character table

306 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1458, 247);