C₅₉×#13633 #28425742

order	15104
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₅₉ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₅₉×#13633

Subgroups

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

190 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 7552

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

(1,2,3,4)(5,6)(7,18)(8,14)(9,27)(10,31)(11,33)(12,17)(13,21)(15,19)(16,28)(20,40)(22,53)(23,55)(24,43)(25,56)(26,39)(29,59)(30,58)(32,60)(34,48)(35,62)(36,52)(37,45)(38,47)(41,44)(42,50)(46,57)(49,65)(51,61)(54,68)(63,66)(64,67)(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);(5,7,19,44,10,22,25,48,8,20,45,66,26,49,30,52,12,24,47,67,32,54,29,51,11,23,46,62,28,50,9,21)(6,13,27,42,16,35,57,55,33,61,59,68,60,64,38,43,17,36,58,65,39,63,37,40,14,34,56,53,31,41,15,18);(5,8,12,11)(6,14,17,33)(7,20,24,23)(9,25,30,29)(10,26,32,28)(13,34,36,61)(15,37,38,57)(16,31,39,60)(18,40,43,55)(19,45,47,46)(21,48,52,51)(22,49,54,50)(27,56,58,59)(35,41,63,64)(42,53,65,68)(44,66,67,62)

Conjugacy classes & character table

4484 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 6621);