C₁₉×#8521 #18014480

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#8521

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

690 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 1216

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)(22,28,30,25)(23,26)(24,27)(29,34,35,32)(31,33)(36,37)(38,46,49,58)(39,50,52,43)(40,44)(41,45)(42,48,56,53)(47,61)(51,65,67,57)(54,59)(55,62)(60,63)(64,66);(20,22)(21,25)(23,29)(24,30)(26,32)(27,28)(31,35)(33,34)(36,38,41,49)(37,42,45,56)(39,47,52,63)(40,53,54,48)(43,55,50,64)(44,58,59,46)(51,66,67,62)(57,61,65,60);(20,23,24,31)(21,26,27,33)(22,29,30,35)(25,32,28,34)(36,39)(37,43)(38,47)(40,51)(41,52)(42,55)(44,57)(45,50)(46,60)(48,62)(49,63)(53,66)(54,67)(56,64)(58,61)(59,65)

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, 1509);