C₁₉×#7345 #18013304

order	4864
nilpotency class	5
isoclinism	isoclinic class of #1404
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#7345

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	AΣL(1,𝔽₁₆)
commutator	C₂×Q₈	C₂×C₈×C₁₉
Frattini, cosocle	#391	C₂²×C₁₉
prime radical	C₂×C₁₉×2₋¹⁺⁴	C₄
LCS	G ▷ C₂×Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#391 ◁ G

342 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 43
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 43 points:

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

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, 333);