C₁₉×#18997 #18024956

order	4864
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₁₉ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

direct product

C₁₉×#18997

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₉
prime radical	C₂×C₁₉×H(3,𝔽₄)	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ G

1066 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1066 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) = 51
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,31)(22,26,33,41)(23,27,35,44)(24,28,37,45)(29,46,47,32)(30,38,42,36)(34,50,49,39)(40,48,43,51);(21,26)(22,32)(23,35)(27,39)(28,29)(30,43)(31,41)(33,46)(34,51)(36,38)(40,42)(44,50)(45,47)(48,49);(21,27)(22,33)(23,36)(26,39)(28,42)(29,40)(30,45)(31,44)(32,46)(34,48)(35,38)(41,50)(43,47)(49,51);(21,28)(26,29)(27,42)(30,44)(31,45)(39,40)(41,47)(43,50)

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, 11985);