C₁₉×#13568 #18019527

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

Aliases

direct product

C₁₉×#13568

Subgroups

	subgroup	G/H
center	C₄×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂²×C₄×C₁₉
Frattini, cosocle	#124	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

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

normal subgroups only

This group has 498 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) = 35
transitive degree	μ_t(G) = 304

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,27)(22,25,30,32)(24,29,34,35)(26,31,28,33);(20,21)(22,25)(23,32)(24,29)(26,34)(27,33)(28,31)(30,35);(20,22,24,31)(21,25,29,28)(23,30,34,33)(26,27,32,35)

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, 6556);