C₁₉×#7401 #18013360

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7398
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7401

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1343
commutator	C₂²×C₈	C₂×C₄×C₁₉
Frattini, cosocle	#408	C₂²×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#408 ◁ G

234 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 83
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,29,26,30,44,59)(22,32,34,62,36,63,67,77)(23,37,40,57,42,28,51,54)(25,31,45,58,39,53,72,76)(27,47,48,38,49,70,78,71)(33,50,65,56,66,60,79,82)(35,52,68,80,46,69,43,74)(41,61,73,83,64,55,75,81);(20,22,26,36)(21,27,30,49)(23,38,42,71)(24,43,44,68)(25,46,39,35)(28,52,37,69)(29,56,59,82)(31,60,53,50)(32,61,63,55)(33,64,66,41)(34,45,67,72)(40,65,51,79)(47,75,70,73)(48,76,78,58)(54,77,57,62)(74,81,80,83)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 389);