C₁₉×#7360 #18013319

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7360

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#354
commutator	C₄²	C₂×C₈×C₁₉
Frattini, cosocle	#423	C₂²×C₁₉
socle	C₂×C₁₉	#1319
prime radical	C₂³×C₁₉	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#423 ◁ G

164 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 46 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)(20,21,23,28,24,29,38,47,25,30,39,48,40,27,37,45)(22,31,33,49,35,26,41,42,32,43,50,46,34,44,36,51);(20,22,25,32)(21,26)(23,36,39,41)(24,35,40,34)(27,31)(28,46)(29,43)(30,44)(33,37,50,38)(42,45)(47,51)(48,49)

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 348);