C₁₉×#7240 #18013199

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

Aliases

direct product

C₁₉×#7240

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)(20,21,24,30,26,32,47,60,27,33,48,31,41,59,25,49)(22,34,36,61,38,63,70,83,39,64,71,62,65,82,37,72)(23,40,42,73,44,56,43,58,45,74,46,76,75,29,54,55)(28,50,51,78,52,79,77,35,53,66,57,68,80,67,81,69);(20,22,25,37,41,65,48,71,27,39,47,70,26,38,24,36)(21,28,31,51,32,52,49,77,33,53,30,57,59,80,60,81)(23,35,43,67,75,78,42,66,45,69,54,79,44,68,46,50)(29,34,55,62,40,63,73,72,56,64,58,61,74,82,76,83)

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, 228);