C₁₉×#12932 #18018891

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

Aliases

direct product

C₁₉×#12932

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#481
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂×C₁₉	#1853
prime radical	C₂⁴×C₁₉	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×M₁₆×C₁₉ ◁ G

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

normal subgroups only

This group has 402 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)(20,21,24,28,25,30,40,29)(22,31,34,49,33,48,39,45)(23,35,38,27,37,47,50,46)(26,41,43,32,42,36,44,51);(20,22,25,33)(21,26)(23,36,37,41)(24,39,40,34)(27,45)(28,44)(29,43)(30,42)(31,35)(32,50,51,38)(46,49)(47,48);(23,37)(27,46)(31,48)(32,51)(35,47)(36,41)(38,50)(45,49)

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, 5920);