C₁₉×#7354 #18013313

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂×C₈×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7354

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#353
commutator	C₄²	C₂×C₈×C₁₉
Frattini, cosocle	#384	C₂²×C₁₉
prime radical	C₂×C₁₉×#127	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#384 ◁ G

268 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 608

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

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 342);