Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of D128 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C59 · Schur multiplier (H2) = C1 |
| direct product | C59×SD256 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C59 | D64 |
| commutator, Frattini, cosocle | C64 | C22×C59 |
| prime radical | C59×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C59 ◁ C4×C59 ◁ C8×C59 ◁ C16×C59 ◁ C32×C59 ◁ C59×C64 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 187 |
| transitive degree | μt(G) = 7552 |
A minimal faithful permutation representation — 2 generators on 187 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187);(2,64)(3,127)(4,62)(5,125)(6,60)(7,123)(8,58)(9,121)(10,56)(11,119)(12,54)(13,117)(14,52)(15,115)(16,50)(17,113)(18,48)(19,111)(20,46)(21,109)(22,44)(23,107)(24,42)(25,105)(26,40)(27,103)(28,38)(29,101)(30,36)(31,99)(32,34)(33,97)(35,95)(37,93)(39,91)(41,89)(43,87)(45,85)(47,83)(49,81)(51,79)(53,77)(55,75)(57,73)(59,71)(61,69)(63,67)(66,128)(68,126)(70,124)(72,122)(74,120)(76,118)(78,116)(80,114)(82,112)(84,110)(86,108)(88,106)(90,104)(92,102)(94,100)(96,98)3953 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 540);