C₁₉×#13572 #18019531

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

Aliases

direct product

C₁₉×#13572

Subgroups

	subgroup	G/H
center	C₄×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×AΣL(1,𝔽₁₆)
prime radical	C₁₉×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

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

normal subgroups only

This group has 482 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)(20,21,24,30,26,32,40,48)(22,33,36,44,37,46,51,27)(23,28,38,45,25,31,42,29)(34,50,35,49,41,47,39,43);(20,22)(21,27)(23,34)(24,39)(25,41)(26,37)(28,43)(29,46)(30,47)(31,49)(32,44)(33,45)(35,40)(36,38)(42,51)(48,50);(20,23,26,25)(21,28,32,31)(22,34,37,41)(24,38,40,42)(27,43,44,49)(29,30,45,48)(33,50,46,47)(35,51,39,36)

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, 6560);