C₁₉×#7211 #18013170

order	4864
nilpotency class	3
isoclinism	isoclinic class of #1304
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7211

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#95
commutator	C₂³	C₄×C₈×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
socle	C₂²×C₁₉	#336
prime radical	C₂³×C₁₉	C₄×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

200 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 43
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 2 generators on 43 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,25,23,24,26,27)(28,29,32,35,34,37,42,30,31,36,41,40,43,38,33,39);(21,24)(22,26)(28,30,33,38,43,35,41,36,31,39,42,37,34,40,32,29)

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, 199);