C₁₉×#11458 #18017417

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×#11458

Subgroups

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

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

normal subgroups only

This group has 364 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)(20,21,23,28,24,29,34,42,25,27,33,39,35,40,46,49)(22,26,31,37,32,38,45,48,30,36,43,41,44,47,51,50);(20,22)(21,26,27,36)(23,31)(24,32)(25,30)(28,41,39,37)(29,38,40,47)(33,43)(34,45)(35,44)(42,50,49,48)(46,51);(22,30)(26,36)(31,43)(32,44)(37,41)(38,47)(45,51)(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, 4446);