C₁₉×#11438 #18017397

order	4864
nilpotency class	4
isoclinism	isoclinic class of #2127
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#11438

Subgroups

	subgroup	G/H
center	C₄×C₁₉	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₁₉
Frattini, cosocle	C₂×M₁₆	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#323
prime radical	C₂²×#41×C₁₉	C₄
LCS	G ▷ C₂×C₄ ▷ 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)(20,21,24,29,25,28,39,45,26,30,40,47,23,35,37,46)(22,31,34,48,33,27,41,43,32,44,38,50,36,42,49,51);(20,22)(21,27)(23,36)(24,38)(25,33)(26,32)(28,44)(29,43)(30,42)(31,35)(34,40)(37,41)(39,49)(45,50)(46,48)(47,51);(22,32)(27,42)(31,44)(33,36)(34,38)(41,49)(43,51)(48,50)

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, 4426);