C₁₉×#13589 #18019548

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

Aliases

direct product

C₁₉×#13589

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#353
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#353
prime radical	C₁₉×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

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

normal subgroups only

This group has 408 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)(20,21,24,29,25,30,40,47)(22,31,33,46,35,42,49,50)(23,27,36,44,37,28,39,45)(26,38,41,32,43,48,51,34);(20,22)(21,26)(23,32)(24,38)(25,35)(27,31)(28,42)(29,46)(30,43)(33,36)(34,37)(39,49)(40,48)(41,45)(44,51)(47,50);(20,23,25,37)(21,27,30,28)(22,32,35,34)(24,36,40,39)(26,31,43,42)(29,44,47,45)(33,48,49,38)(41,46,51,50)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6577);