C₁₉×#8633 #18014592

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#8633

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#380
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂³×C₁₉	#117
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

320 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 67
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27,24,26,32,34)(22,28,29,35,30,25,31,33)(36,37)(38,47)(39,52,54,59)(40,55)(41,45)(42,46)(43,49)(44,53,60,56)(48,57,64,67)(50,58)(51,62)(61,63,66,65);(36,38,41,50,42,51,40,49)(37,43,55,62,46,58,45,47)(39,48,56,65,54,64,53,63)(44,57,52,66,60,67,59,61);(20,22,24,30)(21,25,26,28)(23,31,32,29)(27,35,34,33)(36,39)(37,44)(38,48)(40,53)(41,56)(42,54)(43,57)(45,59)(46,60)(47,61)(49,63)(50,65)(51,64)(52,55)(58,67)(62,66)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1621);