C₁₉×#32786 #18038745

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

Aliases

direct product

C₁₉×#32786

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

716 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 234 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)(22,24)(23,25)(26,27)(36,37,41,44)(38,45,39,42)(40,43,48,51)(46,50,47,49);(20,22)(21,24)(23,26)(25,27)(28,29)(30,32)(31,34)(33,35)(36,38,41,39)(37,42,44,45)(40,47,48,46)(43,50,51,49);(21,25)(24,27)(29,32)(30,35)(33,34)(36,39,41,38)(37,42,44,45)(40,46,48,47)(43,50,51,49);(22,26)(24,27)(36,40,41,48)(37,43,44,51)(38,46,39,47)(42,49,45,50)

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, 25774);