C₁₉×#12223 #18018182

order	4864
nilpotency class	4
isoclinism	isoclinic class of #12099
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₁₉×#12223

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₉
prime radical	C₂²×D₄×C₁₉	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂×C₄⋊C₄×C₁₉ ◁ G

394 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 74 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)(20,21,25,31)(22,30,35,48)(23,36,38,29)(24,28,42,47)(26,33,40,32)(27,37,43,34)(39,49,51,46)(41,45,50,44);(20,22,25,35)(21,27,31,43)(23,37,38,34)(24,40,42,26)(28,46,47,49)(29,48,36,30)(32,44,33,45)(39,50,51,41);(21,28)(22,24)(23,38)(27,29)(30,33)(31,47)(32,48)(34,50)(35,42)(36,43)(37,41)(39,51)(44,49)(45,46)

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, 5211);