C₁₉×#7526 #18013485

order	4864
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7526

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1404
commutator	#125	C₂×C₄×C₁₉
Frattini, cosocle	#491	C₂²×C₁₉
prime radical	C₂²×C₁₉	AΣL(1,𝔽₁₆)
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄²×C₁₉ ◁ C₁₉×#491 ◁ G

126 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 30 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)(20,21,24,29,25,31,22,32)(23,35,37,45,39,46,51,27)(26,30,34,49,41,50,40,48)(28,36,44,42,47,33,43,38);(20,22,25,24)(21,27,31,45)(23,36,39,33)(26,40,41,34)(28,32,47,29)(30,44,50,43)(35,48,46,49)(37,38,51,42)

Conjugacy classes & character table

361 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 514);