C₁₉×#7488 #18013447

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7488

Subgroups

	subgroup	G/H
center	C₄×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₄²×C₁₉
Frattini, cosocle	#532	C₂²×C₁₉
socle	C₂×C₁₉	#1390
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

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

normal subgroups only

This group has 270 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)(20,21,23,29,26,31,38,45)(22,32,34,44,36,49,50,27)(24,30,37,28,25,39,41,46)(33,43,40,47,51,42,35,48);(20,22,24,35,26,36,25,40)(21,27,30,42,31,44,39,43)(23,33,37,50,38,51,41,34)(28,32,45,48,46,49,29,47)

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, 476);