C₁₉×#12165 #18018124

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12108
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#12165

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
prime radical	C₂²×D₈×C₁₉	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#424 ◁ G

350 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 1216

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)(24,27)(29,35)(30,39)(31,42)(32,33)(36,50)(37,54)(38,49)(40,56)(41,46)(43,47)(44,45)(48,51)(52,59)(53,57);(20,21)(22,24)(23,25)(26,27)(28,29,34,38)(30,36,41,53)(31,43,45,54)(32,37,33,47)(35,44,49,42)(39,51,46,52)(40,57,56,50)(48,58,59,55);(20,22,23,26)(21,24,25,27)(28,30)(29,36)(31,40)(32,46)(33,39)(34,41)(35,48)(37,52)(38,53)(42,55)(43,57)(44,58)(45,56)(47,51)(49,59)(50,54)

Conjugacy classes & character table

589 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5153);