C₁₉×#19058 #18025017

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#19058

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₁₉×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

840 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 222 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)(20,21,24,27)(22,28,29,35)(23,31,32,26)(25,33,34,30)(36,37)(38,48)(39,53)(40,45)(41,46)(42,47)(43,51)(44,55)(49,57)(50,58)(52,61)(54,59)(56,60)(62,64)(63,66)(65,67);(23,32)(26,31)(28,35)(30,33)(36,38)(37,43)(39,49)(40,50)(41,51)(42,52)(44,57)(45,48)(46,61)(47,58)(53,66)(54,63)(55,64)(56,65)(59,67)(60,62);(20,22)(21,25)(23,33)(24,29)(26,35)(27,34)(28,31)(30,32)(36,39,42,56)(37,44,47,59)(38,49,52,65)(40,55,41,54)(43,57,58,67)(45,60,46,53)(48,62,61,66)(50,64,51,63);(36,40,42,41)(37,45,47,46)(38,50,52,51)(39,54,56,55)(43,48,58,61)(44,53,59,60)(49,63,65,64)(57,66,67,62)

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, 12046);