C₁₉×#32768 #18038727

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₁₉×#32768

Subgroups

	subgroup	G/H
center, socle	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
prime radical	C₂⁴×C₁₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₂×C₁₉×#117 ◁ G

736 conjugacy classes of subgroups; only the 206 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 206 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) = 67
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 4 generators on 67 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,24)(22,25,26,27)(28,29)(30,32)(31,33)(34,35)(36,37)(38,42)(39,43)(40,45)(41,44)(46,55)(47,57)(48,56)(49,58)(50,60)(51,59)(52,62)(53,63)(54,61)(64,66)(65,67);(21,24)(22,26)(28,30)(29,32)(31,34)(33,35)(36,38,41,48)(37,42,44,56)(39,49,51,65)(40,52,54,47)(43,58,59,67)(45,62,61,57)(46,53,64,50)(55,63,66,60);(22,26)(25,27)(29,33)(32,35)(36,39,41,51)(37,43,44,59)(38,46,48,64)(40,53,54,50)(42,55,56,66)(45,63,61,60)(47,65,52,49)(57,67,62,58);(20,22,23,26)(21,25,24,27)(30,34)(32,35)(37,44)(42,56)(43,59)(45,61)(55,66)(57,62)(58,67)(60,63)

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, 25756);