C₁₉×#7410 #18013369

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7410
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7410

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	#1344
commutator	C₂²×C₈	C₂×C₄×C₁₉
Frattini, cosocle	#408	C₂²×C₁₉
prime radical	C₂×#41×C₁₉	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#408 ◁ G

166 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 2 generators on 51 points:

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

Conjugacy classes & character table

475 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 398);