C₁₉×#7185 #18013144

order	4864
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7185

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#95
commutator	C₂³	C₄×C₈×C₁₉
Frattini, cosocle	C₂³×C₈	C₂²×C₁₉
socle	C₂²×C₁₉	#336
prime radical	C₂⁴×C₁₉	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂³×C₈×C₁₉ ◁ G

314 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 110 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 — 2 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)(22,26,27,24)(23,25)(28,29,31,35,32,37,42,49,33,36,43,48,44,50,55,58)(30,38,40,51,41,52,54,59,39,34,45,46,53,47,56,57);(20,22)(21,24)(23,27)(25,26)(28,30)(29,34,36,38)(31,40)(32,41)(33,39)(35,46,48,51)(37,47,50,52)(42,54)(43,45)(44,53)(49,57,58,59)(55,56)

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, 173);