C₁₉×#10669 #18016628

order	4864
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#10669

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#399
prime radical	C₂³×C₁₉	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

296 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 106 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)(20,21,24,26,25,29,23,28)(22,30,32,35,33,34,31,27)(36,37,42,46,41,47,40,45)(38,48,56,43,51,59,50,58)(39,44,54,63,55,62,53,61)(49,64,52,67,66,60,65,57);(21,26)(22,31)(23,24)(27,35)(28,29)(32,33)(36,38,40,56,41,51,42,50)(37,43,45,59,47,58,46,48)(39,52,53,66,55,65,54,49)(44,60,61,57,62,64,63,67);(20,22,25,33)(21,27,29,35)(23,32,24,31)(26,34,28,30)(36,39,41,55)(37,44,47,62)(38,49,51,66)(40,53,42,54)(43,57,58,67)(45,61,46,63)(48,60,59,64)(50,65,56,52)

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, 3657);