C₁₉×#18999 #18024958

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

Aliases

direct product

C₁₉×#18999

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂²×D₄×C₁₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

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

normal subgroups only

This group has 850 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) = 1216

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

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, 11987);