C₁₉×#7364 #18013323

order	4864
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7364

Subgroups

	subgroup	G/H
center	C₈×C₁₉	#99
commutator	C₂×C₄	C₂×C₁₆×C₁₉
Frattini, cosocle	C₂×M₃₂	C₂²×C₁₉
socle	C₂×C₁₉	#1322
prime radical	C₂³×C₁₉	M₃₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂×C₈×C₁₉ ◁ C₂×C₁₉×M₃₂ ◁ G

126 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,29,24,31,36,44,25,28,35,41,37,42,48,50,26,30,34,43,38,45,47,51,22,27,32,39,33,40,46,49);(21,27,30,28)(23,34)(29,39,43,41)(31,40,45,42)(32,35)(36,47)(44,49,51,50)(46,48)

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, 352);