C₁₉×#19080 #18025039

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

Aliases

direct product

C₁₉×#19080

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

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

normal subgroups only

This group has 936 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) = 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)(21,26)(22,30)(23,24)(28,29)(31,32)(33,34)(36,37,41,45)(38,46,48,51)(39,44,40,43)(42,49,50,47);(20,21,25,29)(22,27,32,35)(23,28,24,26)(30,34,31,33)(36,38,41,48)(37,42,45,50)(39,47,40,49)(43,46,44,51);(20,22)(21,27)(23,30)(24,31)(25,32)(26,33)(28,34)(29,35)(39,40)(43,44)(46,51)(47,49);(20,23,25,24)(21,28,29,26)(22,31,32,30)(27,33,35,34)(36,39,41,40)(37,43,45,44)(38,47,48,49)(42,46,50,51)

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, 12068);