C₁₉×#13052 #18019011

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#13052

Subgroups

	subgroup	G/H
center	C₄×C₁₉	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
socle	C₂×C₁₉	#1889
prime radical	C₁₉×#3018	C₂
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂×#41×C₁₉ ◁ G

624 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 114 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)(20,21,25,30,23,28,32,41)(22,31,35,47,33,26,40,43)(24,29,38,46,36,27,39,45)(34,44,48,51,37,42,49,50);(20,22,32,40,23,33,25,35)(21,26,41,47,28,31,30,43)(24,37,38,49,36,34,39,48)(27,44,45,51,29,42,46,50);(21,27)(22,33)(24,36)(26,42)(28,29)(30,46)(31,44)(35,40)(38,39)(41,45)(43,51)(47,50)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6040);