C₁₉×#7208 #18013167

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

Aliases

direct product

C₁₉×#7208

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#99
commutator	C₂×C₄	C₄×C₈×C₁₉
Frattini, cosocle	C₂×#131	C₂²×C₁₉
socle	C₂×C₁₉	#1282
prime radical	C₂×#41×C₁₉	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ 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)(20,21,24,30,26,32,45,55,27,31,25,46,44,56,47,72)(22,33,35,57,36,60,64,80,37,59,42,58,65,79,69,81)(23,38,39,53,40,66,67,78,41,29,43,52,68,54,70,77)(28,48,49,62,51,74,75,63,50,61,71,73,76,82,83,34);(20,22,25,35,26,36,47,64,27,37,24,42,44,65,45,69)(21,28,30,49,32,51,55,75,31,50,46,71,56,76,72,83)(23,34,39,61,40,62,67,82,41,63,43,48,68,73,70,74)(29,33,53,58,54,60,78,81,38,59,52,57,66,79,77,80)

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, 196);