C₁₉×#7397 #18013356

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7397

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₉
Frattini, cosocle	#407	C₂²×C₁₉
socle	C₂×C₁₉	#1343
prime radical	C₁₉×#2010	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#407 ◁ G

268 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

589 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 385);