C₁₉×#8225 #18014184

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#8225

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²×C₁₉
Frattini, cosocle	C₂×C₄²	C₂³×C₁₉
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₄³×C₁₉ ◁ G

444 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 4864

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)(20,21,23,22)(24,25)(26,32)(27,35)(28,31)(29,34)(30,36)(33,38)(37,39)(40,41,43,46)(42,48,49,45)(44,52,51,47)(50,53,55,54);(24,26)(25,29,31,37)(27,34)(28,33)(30,32,35,38)(36,39)(40,42,44,50,43,49,51,55)(41,45,47,53,46,48,52,54);(20,22,23,21)(26,33)(29,37)(32,38)(34,39)

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, 1213);