C₁₉×#8327 #18014286

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#8327

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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)(20,21,23,26,24,27,30,33)(22,25,29,32,28,31,34,35)(36,37)(38,46)(39,50,52,43)(40,53)(41,45)(42,48)(44,55)(47,61,63,60)(49,59)(51,58,67,65)(54,57)(56,62,64,66);(20,22)(21,25)(23,29)(24,28)(26,32)(27,31)(30,34)(33,35)(36,38,41,49)(37,42,45,57)(39,47,52,63)(40,54,55,48)(43,56,50,64)(44,59,53,46)(51,66,67,62)(58,61,65,60);(22,28)(25,31)(29,34)(32,35)(36,39,41,52)(37,43,45,50)(38,47,49,63)(40,51,55,67)(42,56,57,64)(44,58,53,65)(46,60,59,61)(48,62,54,66)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1315);