C₁₉×#12800 #18018759

order	4864
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#12800

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×D₄	C₂³×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×D₄×C₁₉ ◁ G

524 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 39
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 39 points:

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

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, 5788);