C₁₉×#7406 #18013365

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7406

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#353
commutator	C₂×M₁₆	C₂×C₄×C₁₉
Frattini, cosocle	#408	C₂²×C₁₉
socle	C₂×C₁₉	#1343
prime radical	C₁₉×#2017	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#408 ◁ G

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

normal subgroups only

This group has 244 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)(22,32,34,61)(23,28,39,52)(25,45,44,75)(26,30,43,58)(27,47,48,37)(31,59,57,38)(33,50,63,55)(35,68,67,83)(36,62,66,81)(40,54,72,56)(41,73,71,82)(42,74,46,51)(49,76,77,69)(53,79,78,70)(60,65,80,64);(20,22)(21,27)(23,37)(24,42)(25,46)(26,36)(28,51)(29,55)(30,49)(31,60)(32,53)(33,41)(34,59)(35,38)(39,65)(40,69)(43,67)(44,66)(45,50)(47,71)(48,57)(52,61)(54,68)(56,81)(58,80)(62,73)(63,72)(64,70)(74,82)(75,77)(76,79)(78,83)

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, 394);