C₁₉×#7252 #18013211

order	4864
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7252

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	#99
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂×#117	C₂²×C₁₉
socle	C₂²×C₁₉	#342
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

344 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 102 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)(20,21)(22,24)(23,26)(25,27)(28,29,32,36,33,37,45,52)(30,38,40,53,41,49,54,58)(31,35,43,50,44,51,56,59)(34,46,47,42,48,55,57,39);(21,24,26,25)(28,30)(29,34)(31,42,44,39)(32,40)(33,41)(35,49,51,38)(36,47)(37,48)(43,55,56,46)(45,54)(50,58,59,53)(52,57)

Conjugacy classes & character table

874 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 240);