C₁₉×#7207 #18013166

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7207

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#99
commutator	C₂³	C₄×C₈×C₁₉
Frattini, cosocle	C₂×#131	C₂²×C₁₉
socle	C₂×C₁₉	#1282
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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)(20,21,24,30,26,33,46,57,27,32,25,31,45,56,47,58)(22,34,36,59,38,62,67,81,39,61,37,60,66,80,68,82)(23,40,41,55,42,69,70,79,43,29,44,52,71,53,72,77)(28,48,49,73,51,74,75,35,50,63,54,64,76,83,78,65);(20,22,25,37,26,38,47,68,27,39,24,36,45,66,46,67)(21,28,31,49,33,51,58,75,32,50,30,54,56,76,57,78)(23,35,41,63,42,64,70,83,43,65,44,48,71,73,72,74)(29,34,52,60,53,62,77,82,40,61,55,59,69,80,79,81)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 195);