C₁₉×#7336 #18013295

order	4864
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7336

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#99
commutator	C₂³	C₂×C₁₆×C₁₉
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₉
socle	C₂×C₁₉	#1314
prime radical	C₂⁴×C₁₉	C₁₆
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₉×M₃₂ ◁ G

170 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 54 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)(20,21,22,27,23,28,31,39,24,29,32,40,33,41,44,49,25,26,30,36,34,37,42,46,35,38,43,47,45,48,50,51);(21,26)(28,37)(29,38)(41,48)

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, 324);