C₁₉×#33564 #18039523

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₁₉×#33564

Subgroups

	subgroup	G/H
center	C₄×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×Q₈	C₂⁴×C₁₉
socle	C₂×C₁₉	C₂×B(4,𝔽₂)
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×#41×C₁₉ ◁ G

1112 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 218 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 — 4 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)(22,32)(23,29)(24,37)(25,28)(26,31)(27,33)(30,41)(34,48)(35,43)(36,46)(38,45)(39,47)(40,49)(42,50)(44,51);(20,22)(21,27)(23,33)(24,38)(25,40)(26,35)(28,42)(29,43)(30,44)(31,45)(32,37)(34,41)(36,51)(39,50)(46,49)(47,48);(21,28)(25,41)(27,42)(29,46)(30,31)(32,48)(34,40)(36,39)(37,47)(43,49)(44,45)(50,51);(20,23,26,24)(21,29,31,37)(22,33,35,38)(25,36,41,39)(27,43,45,32)(28,46,30,47)(34,50,40,51)(42,49,44,48)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 26552);