C₅₉×#7084 #28419193

order	15104
nilpotency class	4
isoclinism	isoclinic class of #7076
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₉×#7084

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₉	#327
commutator	C₂²×C₈	C₂×C₄×C₅₉
Frattini, cosocle	C₂×C₂²⋊C₈	C₂²×C₅₉
prime radical	C₅₉×#1999	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×C₂²⋊C₈×C₅₉ ◁ G

316 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 83 points:

(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,18,13,22)(11,15,19,16,21,24,20,23)(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);(2,5)(6,7)(10,15)(11,19)(12,13)(16,17)(18,24)(20,21)(22,23)

Conjugacy classes & character table

2006 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 72);