C₁₉×#12891 #18018850

order	4864
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₁₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₁₉×#12891

Subgroups

	subgroup	G/H
center	C₂×C₄×C₁₉	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₁₉
Frattini, cosocle	C₂²×C₈	C₂³×C₁₉
socle	C₂²×C₁₉	C₂×#107
prime radical	C₂³×C₁₉	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₉ ◁ C₂×C₄²×C₁₉ ◁ C₂×C₄×C₈×C₁₉ ◁ G

282 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 59
transitive degree	μ_t(G) = 2432

A minimal faithful permutation representation — 3 generators on 59 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)(28,29,34,40,35,42,48,41)(30,37,44,55,45,56,46,51)(31,36,43,52,47,54,59,53)(32,39,33,38,49,57,50,58);(21,24)(25,27)(28,30,33,43,34,44,49,47,35,45,50,59,48,46,32,31)(29,36,39,51,41,53,58,56,42,54,57,55,40,52,38,37);(20,22,23,26)(21,25,24,27)(28,31,32,46,48,59,50,45,35,47,49,44,34,43,33,30)(29,37,38,52,40,55,57,54,42,56,58,53,41,51,39,36)

Conjugacy classes & character table

1444 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4864, 5879);