C₁₉×#12365 #18018324

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12364
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12365

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1629
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₁₉×#3291	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₄×C₈×C₁₉ ◁ G

346 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 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)(22,31)(23,34,36,51)(24,29)(25,39)(26,41)(27,33)(28,32,45,50)(30,38)(35,43,42,48)(37,46)(40,47,49,44);(20,22,24,33)(21,27,29,31)(23,32,36,50)(25,26,37,38)(28,43,45,48)(30,46,41,39)(34,44,51,47)(35,40,42,49);(20,23)(21,28)(22,32)(24,36)(25,40)(26,42)(27,43)(29,45)(30,47)(31,48)(33,50)(34,46)(35,38)(37,49)(39,51)(41,44)

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, 5353);