C₁₉×#13611 #18019570

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13611

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#2127
commutator	C₄²	C₂²×C₄×C₁₉
Frattini, cosocle	C₄⋊Q₈	C₂³×C₁₉
prime radical	C₁₉×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#536 ◁ G

316 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 86 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)(20,21,23,29,25,30,36,44)(22,27,32,40,31,41,45,51)(24,26,34,39,37,28,35,43)(33,38,48,49,46,42,47,50);(21,26)(22,31)(23,34)(24,37)(27,38)(28,30)(32,47)(35,36)(39,43)(41,42)(45,48)(49,50);(20,22)(21,27)(23,32)(24,33)(25,31)(26,38)(28,42)(29,40)(30,41)(34,48)(35,47)(36,45)(37,46)(39,49)(43,50)(44,51)

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6599);