C₁₉×#13051 #18019010

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

Aliases

direct product

C₁₉×#13051

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

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

normal subgroups only

This group has 632 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)(22,31,35,47,33,26,40,44)(24,29,38,46,36,27,39,45)(34,42,48,50,37,43,49,51);(20,22,32,40,23,33,25,35)(21,26,41,47,28,31,30,44)(24,37,38,49,36,34,39,48)(27,42,45,50,29,43,46,51);(21,27)(24,36)(26,42)(28,29)(30,46)(31,43)(34,37)(38,39)(41,45)(44,51)(47,50)(48,49)

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, 6039);