C₁₉×#13566 #18019525

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13566

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₉
Frattini, cosocle	#124	C₂³×C₁₉
prime radical	C₁₉×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

478 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 90 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)(20,21,24,30,26,31,39,45)(22,32,35,43,37,49,51,27)(23,28,38,29,25,40,41,46)(33,47,50,44,36,48,34,42);(20,22)(21,27)(23,36)(24,34)(25,33)(26,37)(28,44)(29,32)(30,47)(31,43)(35,41)(38,51)(39,50)(40,42)(45,48)(46,49);(20,23,26,25)(21,28,31,40)(22,33,37,36)(24,38,39,41)(27,42,43,44)(29,45,46,30)(32,47,49,48)(34,35,50,51)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6554);