C₁₉×#12924 #18018883

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#12924

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#447
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1852
prime radical	C₂⁴×C₁₉	Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

386 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 146 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)(20,21,24,29,26,30,25,40)(22,31,34,48,33,47,41,45)(23,35,38,46,37,50,39,28)(27,36,43,49,42,51,44,32);(20,22,24,34,26,33,25,41)(21,27,29,43,30,42,40,44)(23,36,38,49,37,51,39,32)(28,45,35,31,46,48,50,47);(23,37)(28,46)(31,47)(32,49)(35,50)(36,51)(38,39)(45,48)

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, 5912);