C₁₉×#7374 #18013333

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

Aliases

direct product

C₁₉×#7374

Subgroups

	subgroup	G/H
center	C₄×C₁₉	#353
commutator	C₄²	C₂×C₈×C₁₉
Frattini, cosocle	#423	C₂²×C₁₉
socle	C₂×C₁₉	#1325
prime radical	C₂³×C₁₉	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#423 ◁ G

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

normal subgroups only

This group has 148 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)(20,21,23,29,25,30,39,46,26,31,38,47,24,28,40,44)(22,32,34,48,36,42,50,51,33,43,49,45,35,27,37,41);(20,22,24,35,26,33,25,36)(21,27,28,43,31,42,30,32)(23,37,39,34,38,50,40,49)(29,45,46,41,47,48,44,51)

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, 362);