C₅₉×#13656 #28425765

order	15104
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₉×#13656

Subgroups

	subgroup	G/H
center	C₄×C₅₉	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₉
socle	C₂×C₅₉	C₄×D₁₆
prime radical	C₅₉×#2262	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₅₉ ◁ C₂×C₈×C₅₉ ◁ C₄×C₈×C₅₉ ◁ C₄×C₁₆×C₅₉ ◁ G

184 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 123
transitive degree	μ_t(G) = 3776

A minimal faithful permutation representation — 3 generators on 123 points:

(1,2,6,12,8,15,30,14)(3,16,19,47,21,48,20,54)(4,13,23,39,24,10,7,31)(5,25,27,57,29,38,62,58)(9,33,35,53,37,52,45,18)(11,22,41,55,43,56,42,28)(17,49,50,64,51,46,32,40)(26,44,59,63,61,34,60,36)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3)(2,9)(4,17)(5,26)(6,19)(7,32)(8,21)(10,34)(11,40)(12,35)(13,44)(14,45)(15,37)(16,25)(18,28)(20,30)(22,33)(23,50)(24,51)(27,59)(29,61)(31,36)(38,48)(39,63)(41,49)(42,46)(43,64)(47,57)(52,56)(53,55)(54,58)(60,62);(1,4,6,23,8,24,30,7)(2,10,12,31,15,13,14,39)(3,17,19,50,21,51,20,32)(5,22,27,55,29,56,62,28)(9,34,35,36,37,44,45,63)(11,38,41,58,43,25,42,57)(16,46,47,40,48,49,54,64)(18,26,33,59,53,61,52,60)

Conjugacy classes & character table

3068 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 6644);