C₁₉×#12858 #18018817

order	4864
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₁₉×#12858

Subgroups

	subgroup	G/H
center, socle	C₂²×C₁₉	C₂×#99
commutator	C₂²×C₄	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×D₄	C₂³×C₁₉
prime radical	C₂³×D₄×C₁₉	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉ ◁ C₂⁴×C₁₉ ◁ C₂³×D₄×C₁₉ ◁ G

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

normal subgroups only

This group has 786 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) = 43
transitive degree	μ_t(G) = 608

A minimal faithful permutation representation — 3 generators on 43 points:

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

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, 5846);