C₁₉×#7326 #18013285

order	4864
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#7326

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	#342
commutator	C₂²×C₄	C₄²×C₁₉
Frattini, cosocle	C₂²×C₄⋊C₄	C₂²×C₁₉
prime radical	C₂⁴×C₁₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂⁴×C₁₉ ◁ C₂²×C₄⋊C₄×C₁₉ ◁ G

374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 608

A minimal faithful permutation representation — 2 generators on 43 points:

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

Conjugacy classes & character table

646 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 314);