C₁₉×#7239 #18013198

order	4864
nilpotency class	3
isoclinism	isoclinic class of #7225
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₁₉×#7239

Subgroups

	subgroup	G/H
center	C₄×C₁₉	H(3,ℤ/4)
commutator	C₂×C₈	C₄²×C₁₉
Frattini, cosocle	#405	C₂²×C₁₉
socle	C₂×C₁₉	#1290
prime radical	C₂³×C₁₉	#108
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₁₉ ◁ C₂×C₈×C₁₉ ◁ G

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

normal subgroups only

This group has 148 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) = 83
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 83 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,30,26,31,46,59,27,32,47,60,49,55,25,48)(22,33,35,62,37,64,36,70,38,65,44,63,71,81,72,82)(23,39,40,73,42,58,41,57,43,75,45,74,76,29,54,56)(28,50,51,67,52,78,79,68,53,69,80,83,61,66,77,34);(20,22,25,36,49,71,47,35,27,38,46,72,26,37,24,44)(21,28,30,51,31,52,59,79,32,53,60,80,55,61,48,77)(23,34,41,66,76,83,40,69,43,68,54,78,42,67,45,50)(29,33,57,63,39,64,74,82,58,65,56,62,75,81,73,70)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 227);