C₁₉×#12445 #18018404

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12416
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12445

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#459
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₂×D₈×C₁₉	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₈²×C₁₉ ◁ G

234 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 2432

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

Conjugacy classes & character table

817 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5433);