C₁₉×#7115 #18013074

order	4864
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7115

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#329
commutator	C₄²	C₂×C₈×C₁₉
Frattini, cosocle	#376	C₂²×C₁₉
socle	C₂×C₁₉	#1270
prime radical	C₁₉×#532	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#376 ◁ G

248 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 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)(22,30,32,45,33,42,37,51)(23,34,35,49)(24,28)(25,29)(26,31,39,46,41,48,40,47)(27,36,43,50)(38,44);(20,22)(21,26)(23,31)(24,37)(25,33)(27,42)(28,39)(29,41)(30,43)(32,38)(34,45)(35,48)(36,46)(40,44)(47,50)(49,51)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 103);