C₁₉×#12729 #18018688

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12729

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1782
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
prime radical	C₁₉×#535	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₁₉×#519 ◁ G

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

normal subgroups only

This group has 690 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)(20,21,24,29,25,30,37,45)(22,31,34,46,33,27,41,43)(23,26,36,40,35,28,38,44)(32,42,48,51,47,39,49,50);(21,26)(23,35)(24,37)(27,39)(28,30)(29,40)(31,42)(32,47)(34,41)(43,50)(44,45)(46,51);(20,22)(21,27,30,31)(23,32)(24,34)(25,33)(26,39,28,42)(29,43,45,46)(35,47)(36,48)(37,41)(38,49)(40,50,44,51)

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5717);