C₅₉×#13695 #28425804

order	15104
nilpotency class	5
isoclinism	isoclinic class of #13694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₉ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₅₉×#13695

Subgroups

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

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

normal subgroups only

This group has 194 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) = 127
transitive degree	μ_t(G) = 7552

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

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

Conjugacy classes & character table

4130 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(15104, 6683);