C₁₉×#13054 #18019013

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

Aliases

direct product

C₁₉×#13054

Subgroups

	subgroup	G/H
center	C₄×C₁₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×#41	C₂³×C₁₉
socle	C₂×C₁₉	#1889
prime radical	C₁₉×#3018	C₂
LCS	G ▷ C₂×Q₈ ▷ 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)(22,31)(23,28,32,41)(24,29)(26,36)(27,39)(33,46)(34,43)(35,48,38,51)(37,45)(40,47)(42,49,44,50);(20,22,32,34,25,33,23,36)(21,26,41,31,30,43,28,46)(24,38,40,50,39,35,37,49)(27,44,45,51,29,42,47,48);(21,27)(22,33)(24,39)(26,42)(28,47)(29,30)(31,48)(34,36)(37,40)(41,45)(43,44)(46,51)

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, 6042);