C₁₉×#7192 #18013151

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7192

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂³	C₄×C₈×C₁₉
Frattini, cosocle	C₂²×M₁₆	C₂²×C₁₉
socle	C₂²×C₁₉	#324
prime radical	C₂⁴×C₁₉	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

310 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,27,33,29,35,43,51,30,32,41,48,28,34,42,50)(26,36,37,49,39,46,53,54,40,31,44,45,38,47,52,55);(20,21,22,23)(24,26,28,38,30,40,29,39)(25,31,34,46,32,36,35,47)(27,37,42,52,41,44,43,53)(33,49,50,55,48,45,51,54)

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, 180);