C₁₉×#12047 #18018006

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12047

Subgroups

	subgroup	G/H
center	C₈×C₁₉	C₂×D₈
commutator	C₈	C₂²×C₈×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂×C₁₉	C₈×D₈
prime radical	C₁₉×#508	C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈×C₁₉ ◁ C₂×C₁₆×C₁₉ ◁ C₄×C₁₆×C₁₉ ◁ G

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

normal subgroups only

This group has 212 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)(20,21,25,31,27,32,44,30,28,33,24,43,26,22,34,36)(23,37,39,51,41,35,49,50,42,29,38,46,40,47,48,45);(20,22)(21,27)(23,35)(24,30)(25,36)(26,33)(28,32)(29,41)(31,44)(34,43)(37,40)(38,45)(39,50)(42,47)(46,49)(48,51);(20,23,25,39,27,41,44,49,28,42,24,38,26,40,34,48)(21,29,31,46,32,47,30,45,33,37,43,51,22,35,36,50)

Conjugacy classes & character table

1216 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5035);