C₁₉×#13610 #18019569

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#13610

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#2127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#127	C₂³×C₁₉
prime radical	C₁₉×#535	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#535 ◁ G

336 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 86 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)(22,31,33,26)(23,27)(24,29,39,28)(25,30)(32,46,47,40)(34,41,38,42)(35,48,49,51)(36,44)(37,43,50,45);(20,22,25,34)(21,26,30,42)(23,35,36,32)(24,38,39,33)(27,40,44,51)(28,41,29,31)(37,47,50,49)(43,46,45,48);(20,23)(21,27)(22,32)(24,37)(25,36)(26,40)(28,45)(29,43)(30,44)(31,46)(33,47)(34,35)(38,49)(39,50)(41,48)(42,51)

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, 6598);