C₁₉×#7221 #18013180

order	4864
nilpotency class	4
isoclinism	isoclinic class of #7221
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₁₉×#7221

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#340
commutator	C₂×C₈	C₄²×C₁₉
Frattini, cosocle	C₂×C₄×C₈	C₂²×C₁₉
prime radical	C₂³×C₁₉	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

206 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 51
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 51 points:

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

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, 209);