C₁₉×#8732 #18014691

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#8732

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#381
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#376 ◁ G

404 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 1216

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

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1720);