C₁₉×#13565 #18019524

order	4864
nilpotency class	5
isoclinism	isoclinic class of #13561
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₁₉×#13565

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂×AΣL(1,𝔽₁₆)
commutator	C₂×D₄	C₂²×C₄×C₁₉
Frattini, cosocle	#124	C₂³×C₁₉
prime radical	C₂×C₁₉×H(5,𝔽₂)	C₄
LCS	G ▷ C₂×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×C₄×C₁₉ ◁ C₂×C₁₉×#124 ◁ G

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

normal subgroups only

This group has 496 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) = 608

A minimal faithful permutation representation — 3 generators on 51 points:

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

Conjugacy classes & character table

532 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6553);