C₁₉×#13583 #18019542

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

Aliases

direct product

C₁₉×#13583

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#353
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#127	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

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

normal subgroups only

This group has 416 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)(24,29,38,28)(25,30)(32,45,46,39)(34,40,37,41)(35,43,49,42)(36,44)(47,50,48,51);(20,22)(21,26)(23,32)(24,37)(25,34)(27,39)(28,40)(29,31)(30,41)(33,38)(35,48)(36,47)(42,50)(43,45)(44,51)(46,49);(20,23,25,36)(21,27,30,44)(22,32,34,47)(24,35,38,49)(26,39,41,51)(28,42,29,43)(31,45,40,50)(33,46,37,48)

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, 6571);