C₁₉×#13604 #18019563

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13604

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×#354
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#124	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

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

normal subgroups only

This group has 322 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)(22,32,34,45)(23,28)(24,39)(25,30,41,46)(26,31)(27,33,43,50)(29,40,48,38)(35,42,36,47)(37,49,51,44);(20,22,26,36)(21,27)(23,37,24,33)(25,34,41,35)(28,45)(29,47)(30,49)(31,44)(32,48)(38,51,40,50)(39,42)(43,46);(20,23,26,24)(21,28,31,39)(22,33,36,37)(25,38,41,40)(27,42,44,45)(29,46,48,30)(32,43,47,49)(34,50,35,51)

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, 6592);