C₁₉×#19065 #18025024

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

Aliases

direct product

C₁₉×#19065

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#3548	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

920 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 2432

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)(20,21)(22,30)(23,27)(24,28)(25,29)(26,32)(31,34)(33,35)(36,37,41,44)(38,45,47,50)(39,43,40,48)(42,49,51,46);(20,22)(21,26)(23,31)(24,32)(25,33)(27,30)(28,35)(29,34)(36,38,41,47)(37,42,44,51)(39,46,40,49)(43,50,48,45);(20,22)(21,26)(23,32)(24,31)(25,33)(27,35)(28,30)(29,34)(39,40)(43,48)(45,50)(46,49);(20,23,25,24)(21,27,29,28)(22,31,33,32)(26,30,34,35)(36,39,41,40)(37,43,44,48)(38,46,47,49)(42,50,51,45)

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, 12053);