C₁₉×#8505 #18014464

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

Aliases

direct product

C₁₉×#8505

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#120
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
socle	C₂²×C₁₉	#379
prime radical	C₁₉×#3531	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

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

normal subgroups only

This group has 648 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)(21,25,27,30)(26,34,35,33)(36,37)(38,46,49,42)(39,50,51,61)(40,45)(41,44)(43,52,58,54)(47,63)(48,60,53,57)(55,59)(56,65)(62,66)(64,67);(20,21)(22,26)(23,30)(24,27)(25,32)(28,33)(29,35)(31,34)(36,38)(37,42)(39,47)(40,53)(41,49)(43,56)(44,46)(45,60)(48,55)(50,63)(51,64)(52,66)(54,65)(57,59)(58,62)(61,67);(20,22,24,29)(21,26,27,35)(23,31,32,28)(25,33,30,34)(36,39)(37,43)(38,47)(40,54)(41,51)(42,56)(44,58)(45,61)(46,62)(48,66)(49,64)(50,59)(52,55)(53,65)(57,63)(60,67)

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, 1493);