C₁₉×#11911 #18017870

order	4864
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#11911

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#329
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#97	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#329
prime radical	C₁₉×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#97 ◁ G

318 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 generators on 51 points:

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

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 4899);