C₁₉×#7270 #18013229

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7270

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#331
commutator	C₄²	C₄²×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
socle	C₂×C₁₉	#1296
prime radical	C₂³×C₁₉	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

208 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 114 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 — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,28,26,30,25,40)(22,31,33,42,35,43,32,27)(24,37,36,44,39,50,38,29)(34,48,47,51,49,41,46,45);(20,22,24,34,25,32,38,46,26,35,39,49,23,33,36,47)(21,27,29,41,28,31,37,45,30,42,44,48,40,43,50,51)

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, 258);