C₁₉×#13596 #18019555

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13595
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#13596

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×#354
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

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

normal subgroups only

This group has 326 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)(22,31,34,42)(23,27)(24,29,39,28)(25,30)(26,33,41,38)(32,47,48,43)(35,49,50,40)(36,45,51,44)(37,46);(20,22,25,33)(21,26)(23,35,37,32)(24,38,39,34)(27,43)(28,31)(29,41)(30,42)(36,48,51,50)(40,46)(44,49)(45,47);(20,23,25,37)(21,27,30,46)(22,32,33,35)(24,36,39,51)(26,40,42,43)(28,44,29,45)(31,47,41,49)(34,48,38,50)

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, 6584);