C₁₉×#12213 #18018172

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

Aliases

direct product

C₁₉×#12213

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

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

normal subgroups only

This group has 380 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) = 43
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 43 points:

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

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, 5201);