C₁₉×#7228 #18013187

order	4864
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7228

Subgroups

	subgroup	G/H
center	C₄×C₁₉	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₉
Frattini, cosocle	C₂×#131	C₂²×C₁₉
socle	C₂×C₁₉	#1289
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ G

252 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 86 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)(22,32)(23,37,39,63)(24,42)(25,30)(26,31)(27,33,45,57)(28,41,48,65)(29,50)(34,60)(35,54)(36,55)(38,52)(40,67,66,49)(43,51)(44,70)(46,73,71,81)(47,59,72,78)(53,74)(56,77)(58,80,79,69)(61,76)(62,82)(64,68,83,75);(20,22,24,34,38,56,50,74,26,36,44,62,25,35,43,61)(21,27,29,46,30,47,42,69,31,45,51,71,52,72,70,80)(23,33,40,58,65,78,75,73,39,57,66,79,41,59,68,81)(28,32,49,53,37,54,64,60,48,55,67,76,63,77,83,82)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 216);