C₁₉×#13594 #18019553

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

Aliases

direct product

C₁₉×#13594

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#354
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	#127	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

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

normal subgroups only

This group has 326 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)(22,31,34,41)(23,27)(24,29,38,28)(25,30)(26,33,40,37)(32,45,47,51)(35,43,49,42)(36,44)(39,46,50,48);(20,22,25,33)(21,26)(23,32,36,46)(24,37,38,34)(27,39)(28,31)(29,40)(30,41)(35,48,49,47)(42,45)(43,50)(44,51);(20,23,25,36)(21,27,30,44)(22,32,33,46)(24,35,38,49)(26,39,41,51)(28,42,29,43)(31,45,40,50)(34,47,37,48)

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, 6582);