C₁₉×#8306 #18014265

order	4864
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#8306

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₁₉
Frattini, cosocle	C₂³×C₄	C₂³×C₁₉
prime radical	C₂⁴×C₁₉	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂²×C₄²×C₁₉ ◁ G

810 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 810 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 2432

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

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, 1294);