C₁₉×#11483 #18017442

order	4864
nilpotency class	4
isoclinism	isoclinic class of AΣL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#11483

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#99
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
socle	C₂×C₁₉	#1459
prime radical	C₁₉×#525	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₉×#131 ◁ G

382 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 3 generators on 51 points:

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

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, 4471);