C₁₉×#11922 #18017881

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

Aliases

direct product

C₁₉×#11922

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#331
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄×C₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#331
prime radical	C₁₉×#3147	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

400 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 138 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)(20,21,24,28,37,29,47,45,25,30,39,46,23,27,36,43)(22,31,33,48,32,26,38,41,35,40,50,44,34,42,49,51);(20,22)(21,26)(23,32)(24,38)(25,35)(27,40)(28,44)(29,31)(30,42)(33,47)(34,37)(36,50)(39,49)(41,45)(43,51)(46,48);(20,23,25,37)(21,27,30,29)(22,32,35,34)(24,36,39,47)(26,40,42,31)(28,43,46,45)(33,38,50,49)(41,44,51,48)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 4910);