C₁₉×#21827 #18027786

order	4864
nilpotency class	3
isoclinism	isoclinic class of #17405
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₉×#21827

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂²×C₂²⋊C₄
commutator	C₂³	C₂³×C₄×C₁₉
Frattini, cosocle	C₂⁴	C₂⁴×C₁₉
prime radical	C₂⁵×C₁₉	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₂²⋊C₄×C₁₉ ◁ G

1130 conjugacy classes of subgroups; only the 366 normal subgroups are shown.

normal subgroups only

This group has 1130 conjugacy classes of subgroups — too many to draw. Only its 366 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) = 608

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

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 14815);