C₅₉×#13653 #28425762

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

Aliases

direct product

C₅₉×#13653

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂²×C₅₉	C₂×D₁₆
commutator	C₁₆	C₂²×C₄×C₅₉
Frattini, cosocle	C₂×C₁₆	C₂³×C₅₉
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₉ ◁ C₄²×C₅₉ ◁ C₄×C₈×C₅₉ ◁ C₄×C₁₆×C₅₉ ◁ G

188 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 127
transitive degree	μ_t(G) = 15104

A minimal faithful permutation representation — 3 generators on 127 points:

(1,2,4,3)(5,6)(7,17)(8,22)(9,28)(10,32)(11,13)(12,18)(14,25)(15,24)(16,30)(19,49)(20,52)(21,44)(23,41)(26,60)(27,43)(29,39)(31,62)(33,45)(34,48)(35,56)(36,55)(37,46)(38,50)(40,67)(42,47)(51,68)(53,58)(54,59)(57,66)(61,64)(63,65)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127);(5,7,11,21)(6,12,13,34)(8,23,27,57)(9,29,31,63)(10,33,30,20)(14,39,37,65)(15,41,38,66)(16,42,32,36)(17,43,44,22)(18,46,48,25)(19,50,51,24)(26,61,59,56)(28,35,62,64)(40,68,53,49)(45,54,52,60)(47,58,55,67);(1,3,4,2)(6,13)(12,34)(14,37)(15,38)(16,32)(17,44)(22,43)(28,62)(35,64)(36,42)(39,65)(40,53)(41,66)(45,52)(49,68)(54,60)

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, 6641);