C₁₉×#7347 #18013306

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7347

Subgroups

	subgroup	G/H
center	C₄×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₁₉
Frattini, cosocle	#408	C₂²×C₁₉
socle	C₂×C₁₉	#1316
prime radical	C₁₉×#585	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#408 ◁ G

318 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 46 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) = 83
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,30,26,31,45,63,27,32,46,64,25,47,50,62)(22,33,35,65,36,66,70,82,37,57,71,79,48,78,43,75)(23,38,40,72,41,29,56,58,42,59,44,76,49,60,74,83)(28,51,52,34,54,67,61,69,55,77,80,39,53,73,81,68);(20,22)(21,28)(23,39)(24,43)(25,48)(26,36)(27,37)(29,57)(30,61)(31,54)(32,55)(33,60)(34,42)(35,45)(38,66)(40,51)(41,68)(44,77)(46,70)(47,53)(49,69)(50,71)(52,62)(56,67)(58,82)(59,78)(63,80)(64,81)(65,72)(73,74)(75,83)(76,79)

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, 335);