C₁₉×#12299 #18018258

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12299

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1613
commutator, Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×M₁₆×C₁₉ ◁ G

324 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 70 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)(20,21,24,38,25,29,40,28)(22,30,33,43,32,46,39,47)(23,34,37,27,36,50,51,44)(26,35,45,31,42,48,41,49);(20,22,25,32)(21,26,29,42)(23,35,36,48)(24,39,40,33)(27,43,44,47)(28,45,38,41)(30,34,46,50)(31,37,49,51);(27,44)(30,46)(34,50)(43,47)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5287);