C₁₉×#8527 #18014486

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8516
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#8527

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#379
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

656 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 142 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)(20,21,24,29,25,28,23,27)(22,30,31,35,32,26,33,34)(36,37)(38,46,49,58)(39,50,52,43)(40,44)(41,45)(42,48,56,53)(47,61)(51,65,67,57)(54,59)(55,62)(60,63)(64,66);(20,22,25,32)(21,26,28,30)(23,31,24,33)(27,34,29,35)(36,38)(37,42)(39,47)(40,53)(41,49)(43,55)(44,58)(45,56)(46,59)(48,54)(50,64)(51,66)(52,63)(57,61)(60,65)(62,67);(20,23,25,24)(21,27,28,29)(22,31,32,33)(26,34,30,35)(36,39,41,52)(37,43,45,50)(38,47,49,63)(40,51,54,67)(42,55,56,64)(44,57,59,65)(46,60,58,61)(48,62,53,66)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 1515);