C₁₉×#12021 #18017980

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#12021

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#106
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
prime radical	C₂³×C₁₉	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

216 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 2432

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

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5009);