C₁₉×#7505 #18013464

order	4864
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7505

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#354
commutator	C₄²	C₄²×C₁₉
Frattini, cosocle	C₂×C₄⋊Q₈	C₂²×C₁₉
prime radical	C₂³×C₁₉	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×C₄⋊Q₈ ◁ G

260 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 55 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25,27,32,29,34,41,51)(26,35,37,52,36,47,53,49)(28,33,42,50,43,31,40,48)(30,44,45,38,46,39,55,54);(20,21,22,23)(24,26,28,38,29,36,43,54)(25,30,33,35,34,46,31,47)(27,39,40,37,41,44,42,53)(32,49,48,55,51,52,50,45)

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, 493);