C₁₉×#8782 #18014741

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#8782

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#117
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#382
prime radical	C₂³×C₁₉	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₁₉×#376 ◁ G

262 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 67
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 67 points:

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

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, 1770);