C₁₉×#32816 #18038775

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₄³

Aliases

direct product

C₁₉×#32816

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₉ ◁ C₁₉×#376 ◁ G

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

normal subgroups only

This group has 662 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) = 4864

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)(28,29,31,33)(30,32,34,35)(36,37)(38,42)(39,49)(40,44)(41,45)(43,51)(46,64)(47,57)(48,58)(50,67)(52,63)(53,61)(54,62)(55,60)(56,65)(59,66);(21,24)(22,26)(28,30)(29,32)(31,34)(33,35)(36,38,40,47)(37,42,44,57)(39,50,51,66)(41,54,53,48)(43,59,49,67)(45,62,61,58)(46,55,65,52)(56,63,64,60);(22,26)(25,27)(29,33)(32,35)(36,39,40,51)(37,43,44,49)(38,46,47,65)(41,55,53,52)(42,56,57,64)(45,63,61,60)(48,66,54,50)(58,67,62,59);(20,22,23,26)(21,25,24,27)(30,34)(32,35)(37,44)(38,47)(43,49)(45,61)(46,65)(48,54)(50,66)(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, 25804);