C₁₉×#7388 #18013347

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

Aliases

direct product

C₁₉×#7388

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#357
commutator	C₁₆	C₂×C₈×C₁₉
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₉
socle	C₂×C₁₉	#1331
prime radical	C₁₉×#508	C₄
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₄×C₈×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

118 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 118 conjugacy classes of subgroups — too many to draw. Only its 54 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)(20,21,24,30,26,31,48,58,27,33,43,63,51,32,25,49)(22,34,36,66,38,68,50,67,40,69,37,75,39,76,71,57)(23,41,44,59,46,77,70,61,47,65,81,60,82,78,45,29)(28,52,53,73,55,80,54,35,56,42,62,72,64,79,83,74);(20,22)(21,28)(23,42)(24,37)(25,50)(26,38)(27,40)(29,57)(30,62)(31,55)(32,64)(33,56)(34,65)(35,70)(36,43)(39,51)(41,69)(44,73)(45,74)(46,79)(47,52)(48,71)(49,54)(53,63)(58,83)(59,66)(60,75)(61,67)(68,78)(72,81)(76,77)(80,82)

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, 376);