C₁₉×#32705 #18038664

order	4864
nilpotency class	3
isoclinism	isoclinic class of #31077
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#32705

Subgroups

	subgroup	G/H
center	C₄×C₁₉	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
socle	C₂×C₁₉	#2804
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ G

536 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,29,26,31,24,30)(22,32,35,27,36,43,34,44)(23,37,41,28,39,47,40,46)(33,48,49,45,50,42,38,51);(20,22)(21,27)(23,38)(24,34)(25,35)(26,36)(28,45)(29,43)(30,32)(31,44)(33,41)(37,48)(39,49)(40,50)(42,47)(46,51);(20,23,26,39)(21,28)(22,33)(24,40,25,41)(27,42,44,48)(29,47)(30,37)(31,46)(32,45,43,51)(34,38)(35,49)(36,50);(22,34,36,35)(23,39)(27,43,44,32)(28,46)(33,49,50,38)(37,47)(40,41)(42,45,48,51)

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, 25693);