C₁₉×#13586 #18019545

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13584
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₉×#13586

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×#353
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#127	C₂³×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

406 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 90 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)(22,31,33,26)(23,27)(24,29,39,28)(25,30)(32,47,48,40)(34,41,38,42)(35,49,50,43)(36,45,51,44)(37,46);(20,22,25,34)(21,26,30,42)(23,35,37,32)(24,38,39,33)(27,43,46,40)(28,41,29,31)(36,48,51,50)(44,47,45,49);(20,23,25,37)(21,27,30,46)(22,32,34,35)(24,36,39,51)(26,40,42,43)(28,44,29,45)(31,47,41,49)(33,48,38,50)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6574);