C₁₉×#11391 #18017350

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#11391

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#120
commutator	C₂⁴	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
socle	C₂²×C₁₉	#379
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₄×C₁₉ ◁ G

842 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 842 conjugacy classes of subgroups — too many to draw. Only its 254 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) = 59
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 59 points:

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

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, 4379);