C₁₉×#13458 #18019417

order	4864
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#13458

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

324 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 55 points:

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

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, 6446);