C₁₉×#12764 #18018723

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₁₉×#12764

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator	C₂×D₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×D₄	C₂³×C₁₉
prime radical	C₂²×D₄×C₁₉	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×D₄×C₁₉ ◁ G

768 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 55
transitive degree	μ_t(G) = 1216

A minimal faithful permutation representation — 3 generators on 55 points:

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

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, 5752);