C₁₉×#12864 #18018823

order	4864
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#12864

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#410
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
socle	C₂×C₁₉	#1820
prime radical	C₂×#41×C₁₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₄²×C₁₉ ◁ C₄×#41×C₁₉ ◁ G

342 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 106 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 — 3 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)(22,31,34,29)(23,35)(24,28)(25,30)(26,33,40,47)(27,32,42,46)(36,39)(37,44)(38,43)(41,45,50,51)(48,49);(20,22)(21,26,30,40)(23,32)(24,33)(25,34)(27,39,42,49)(28,31,43,29)(35,41,44,50)(36,45)(37,46)(38,47)(48,51);(20,23,24,36,25,37,38,48)(21,27,28,41,30,42,43,50)(22,32,33,45,34,46,47,51)(26,39,31,44,40,49,29,35)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5852);