C₁₉×#19083 #18025042

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#19083

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

876 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 230 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) = 51
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 4 generators on 51 points:

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

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, 12071);