C₁₉×#7490 #18013449

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7490

Subgroups

	subgroup	G/H
center	C₄×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₄²×C₁₉
Frattini, cosocle	#532	C₂²×C₁₉
socle	C₂×C₁₉	#1390
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

274 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,28)(22,32,34,26)(24,29,39,46)(25,31,41,48)(27,42,30,40)(33,49,36,47)(35,50,38,43)(37,45,51,44);(20,22,24,35,25,37,42,36)(21,26,29,43,31,44,30,49)(23,38,39,51,41,33,40,34)(27,45,28,47,46,32,48,50)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 478);