C₅₉×#11427 #28423536

order	15104
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₉ · Schur multiplier (H₂) = C₂×C₄²

Aliases

direct product

C₅₉×#11427

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#99
commutator	C₂³	C₂²×C₈×C₅₉
Frattini, cosocle	C₂×C₂²⋊C₄	C₂³×C₅₉
prime radical	C₂³×D₄×C₅₉	C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×#117×C₅₉ ◁ G

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

normal subgroups only

This group has 686 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) = 83
transitive degree	μ_t(G) = 1888

A minimal faithful permutation representation — 3 generators on 83 points:

(1,2,3,5,4,6,7,8)(9,10)(11,15)(12,19,20,14)(13,16)(17,23)(18,22,24,21)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83);(10,14)(11,17)(15,22)(16,19)(18,24)(21,23);(9,11,13,17)(10,15,16,23)(12,18,20,24)(14,21,19,22)

Conjugacy classes & character table

2714 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 4415);