C₁₉×#7502 #18013461

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

Aliases

direct product

C₁₉×#7502

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#354
commutator	C₄²	C₄²×C₁₉
Frattini, cosocle	C₂×#124	C₂²×C₁₉
prime radical	C₂²×D₄×C₁₉	D₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#124 ◁ G

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

normal subgroups only

This group has 286 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)(26,36,38,50)(27,41,42,48)(28,34,29,33)(30,35)(31,37,45,53)(32,44)(39,51,40,52)(43,49)(46,54,47,55);(20,21,22,23)(24,26,28,39,30,40,29,38)(25,31,33,45,35,47,34,46)(27,37,43,53,42,55,44,54)(32,36,48,50,49,52,41,51)

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, 490);