C₅₉×#11449 #28423558

order	15104
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₅₉ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₅₉×#11449

Subgroups

	subgroup	G/H
center, socle	C₂²×C₅₉	C₂×#99
commutator	C₂×C₄	C₂²×C₈×C₅₉
Frattini, cosocle	C₂×C₄⋊C₄	C₂³×C₅₉
prime radical	C₂⁴×C₅₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₂²×C₄×C₅₉ ◁ C₂×#117×C₅₉ ◁ G

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

normal subgroups only

This group has 434 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) = 3776

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

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

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, 4437);