C₅₉×#7326 #28419435

order	15104
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄²×C₅₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₉×#7326

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₉	#342
commutator	C₂²×C₄	C₄²×C₅₉
Frattini, cosocle	C₂²×C₄⋊C₄	C₂²×C₅₉
prime radical	C₂⁴×C₅₉	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂⁴×C₅₉ ◁ C₂²×C₄⋊C₄×C₅₉ ◁ G

374 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

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, 314);