C₁₉×#7206 #18013165

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₄×C₈×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7206

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂×C₄	C₄×C₈×C₁₉
Frattini, cosocle	C₂×C₄×Q₈	C₂²×C₁₉
socle	C₂²×C₁₉	#342
prime radical	C₂³×C₁₉	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₄×Q₈×C₁₉ ◁ G

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

normal subgroups only

This group has 284 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)(20,21,23,26,25,27,24,22)(28,29,31,35,32,36,40,43)(30,34,37,42,38,33,39,41);(20,22,24,27,25,26,23,21)(29,33,36,34)(35,42,43,41)

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, 194);