C₁₉×#7429 #18013388

order	4864
nilpotency class	5
isoclinism	isoclinic class of #7412
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7429

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#354
commutator	C₂×C₄²	C₂×C₄×C₁₉
Frattini, cosocle	#391	C₂²×C₁₉
prime radical	C₂³×C₁₉	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₁₉×#391 ◁ G

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

normal subgroups only

This group has 196 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) = 59
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,24,25,26)(28,29,31,36,33,38,44,54)(30,39,40,53,42,50,56,59)(32,35,46,52,47,37,45,55)(34,48,49,41,51,43,58,57);(20,21)(22,26)(23,25)(24,27)(28,30,32,41,33,42,47,57)(29,34,37,50,38,51,35,39)(31,43,45,40,44,48,46,56)(36,53,52,58,54,59,55,49)

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, 417);