C₁₉×#13682 #18019641

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13681
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13682

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₁₉×C₄⋊Q₈ ◁ G

384 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 58 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)(21,27)(22,31)(23,34)(24,37)(29,47)(30,35)(32,51)(33,39)(36,41)(38,49)(42,44)(43,50);(20,21,26,30)(22,28,33,46)(23,35,34,27)(24,38,37,47)(25,29,40,49)(31,50,39,43)(32,45,41,48)(36,44,51,42);(20,22)(21,28)(23,36)(24,39)(25,41)(26,33)(27,42)(29,48)(30,46)(31,37)(32,40)(34,51)(35,44)(38,43)(45,49)(47,50)

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, 6670);