C₁₉×#12774 #18018733

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

Aliases

direct product

C₁₉×#12774

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×H(3,𝔽₄)	C₂
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂³×C₁₉ ◁ C₂²×Q₈×C₁₉ ◁ G

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

normal subgroups only

This group has 932 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) = 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)(20,21)(22,23)(24,25,28,31,29,34,27,33)(26,30,37,38,35,32,36,39);(25,30)(26,35)(27,28)(31,38)(32,34)(33,39);(21,23)(25,31)(27,28)(30,38)(32,39)(33,34)(36,37)

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, 5762);