C₁₉×#19618 #18025577

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₉×#19618

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
socle	C₂²×C₁₉	C₂×#127
prime radical	C₂³×C₁₉	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

690 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 278 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)(22,29)(23,27)(24,33)(25,28)(26,32)(30,35)(31,34)(36,37,40,43)(38,44,47,51)(39,48,45,42)(41,49,50,46);(20,22,24,32,25,30,23,31)(21,26,27,29,28,34,33,35);(22,30)(26,34)(29,35)(31,32)(36,38,45,49,40,47,39,46)(37,41,48,51,43,50,42,44);(20,23,25,24)(21,27,28,33)(22,31,30,32)(26,29,34,35)

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, 12606);