C₁₉×#12066 #18018025

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12066

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₉	#442
commutator	C₈	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₄²×C₁₉ ◁ C₁₉×#322 ◁ G

214 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 59
transitive degree	μ_t(G) = 4864

A minimal faithful permutation representation — 3 generators on 59 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,26,24,22,25,27)(28,29,32,38,34,39,33,36)(30,40,42,57,44,58,43,54)(31,45,48,55,47,53,49,37)(35,50,51,41,52,46,56,59);(28,30,34,44)(29,35,39,52)(31,46,47,50)(32,43,33,42)(36,51,38,56)(37,54,55,57)(40,45,58,53)(41,49,59,48);(20,22,23,27,24,21,25,26)(29,36,39,38)(31,47)(35,51,52,56)(37,45,55,53)(40,54,58,57)(41,59)(46,50)(48,49)

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, 5054);