C₁₉×#11436 #18017395

order	4864
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#11436

Subgroups

	subgroup	G/H
center	C₄×C₁₉	C₂×#99
commutator	C₂³	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#323
prime radical	C₂²×#41×C₁₉	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

504 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 138 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)(20,21,23,28,24,29,35,43,25,27,34,40,36,41,47,50)(22,30,32,42,33,44,46,51,31,26,37,38,45,39,48,49);(20,22)(21,26)(23,32)(24,33)(25,31)(27,30)(28,42)(29,39)(34,37)(35,46)(36,45)(38,40)(41,44)(43,51)(47,48)(49,50);(22,31)(26,30)(32,37)(33,45)(38,42)(39,44)(46,48)(49,51)

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, 4424);