C₁₉×#13057 #18019016

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₁₉×#13057

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	B(4,𝔽₂)
commutator	C₂×Q₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×Q₈	C₂³×C₁₉
prime radical	C₂×C₁₉×#454	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×Q₈×C₁₉ ◁ G

700 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 39
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 39 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,26,30,29,34,27,32)(28,37,36,39,38,33,35,31);(20,21,22,23)(25,30)(26,27)(31,33)(32,34)(35,36)(37,39);(24,26)(25,31)(27,29)(28,36)(30,37)(32,33)(34,39)(35,38)

Conjugacy classes & character table

760 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 6045);