C₁₉×#7467 #18013426

order	4864
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#7467

Subgroups

	subgroup	G/H
center	C₂×C₁₆×C₁₉	D₄
commutator	C₄	C₄×C₁₆×C₁₉
Frattini, cosocle	C₂²×C₁₆	C₂²×C₁₉
socle	C₂²×C₁₉	C₄⋊C₁₆
prime radical	C₂³×C₁₉	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆×C₁₉ ◁ C₂²×C₁₆×C₁₉ ◁ G

190 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 2 generators on 55 points:

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

Conjugacy classes & character table

2128 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 455);