C₁₉×#13598 #18019557

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13598

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#354
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
socle	C₂×C₁₉	C₂×#354
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#532 ◁ G

318 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 318 conjugacy classes of subgroups — too many to draw. Only its 90 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)(20,21,24,29,25,30,40,48)(22,31,34,49,33,42,50,47)(23,27,36,45,37,28,39,46)(26,41,43,35,44,38,51,32);(20,22,25,33)(21,26)(23,32,37,35)(24,38,40,41)(27,42)(28,31)(29,47)(30,44)(34,39,50,36)(43,46)(45,51)(48,49);(20,23,25,37)(21,27,30,28)(22,32,33,35)(24,36,40,39)(26,42,44,31)(29,45,48,46)(34,41,50,38)(43,47,51,49)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6586);