C₁₉×#12225 #18018184

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

Aliases

direct product

C₁₉×#12225

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₁₉
prime radical	C₂³×C₁₉	#120
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂×C₄⋊C₄×C₁₉ ◁ G

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

normal subgroups only

This group has 336 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) = 1216

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)(20,21,24,27)(22,28,30,35)(23,31,33,26)(25,32,34,29)(36,37,41,46)(38,44,39,42)(40,49,50,45)(43,48,51,47);(20,22,24,30)(21,25,27,34)(23,32,33,29)(26,35,31,28)(37,42)(38,39)(40,50)(43,45)(44,46)(49,51);(36,38,41,39)(37,43,46,51)(40,48,50,47)(42,49,44,45)

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, 5213);