C₁₉×#13617 #18019576

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₁₉×#13617

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₁₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₁₉
prime radical	C₂²×C₁₉×D₁₆	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂²×C₈×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

330 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(24,25)(26,32)(27,31)(28,42)(29,44)(30,35)(33,38)(34,39)(36,45)(37,51)(40,48)(41,47)(43,55)(46,52)(49,53)(50,54);(25,31)(26,29)(27,38)(28,39)(32,48)(33,34)(35,47)(36,52)(37,43)(40,53)(41,54)(42,46)(44,45)(49,55)(50,51);(24,26,30,29)(25,32,35,44)(27,36,41,40)(28,37,43,39)(31,45,47,48)(33,49,50,46)(34,42,51,55)(38,53,54,52)

Conjugacy classes & character table

988 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6605);