Finite Group DatabasebetaFinite Group Databasebeta| order | 15104 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C59 · Schur multiplier (H2) = C2 |
| direct product | C59×#7538 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C59 | D32 |
| commutator | C32 | C2×C4×C59 |
| Frattini, cosocle | C2×C32 | C22×C59 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C59 ◁ C2×C4×C59 ◁ C2×C8×C59 ◁ C2×C16×C59 ◁ C2×C32×C59 ◁ G | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 127 |
| transitive degree | μt(G) = 15104 |
A minimal faithful permutation representation — 2 generators on 127 points:
(1,2,3,4)(6,12)(7,17)(8,19)(9,24)(10,28)(13,35)(14,39)(15,42)(16,34)(18,23)(20,50)(21,54)(22,31)(25,59)(26,36)(27,32)(29,40)(30,33)(37,67)(38,66)(41,68)(43,44)(45,52)(46,60)(47,58)(48,49)(51,56)(53,64)(55,57)(61,65)(62,63)(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);(5,6,11,16)(7,18,22,47)(8,23,26,58)(9,27,29,61)(10,30,28,15)(12,31,34,17)(13,36,38,19)(14,40,41,24)(20,51,53,67)(21,55,52,46)(25,60,48,57)(32,64,65,50)(33,45,42,54)(35,62,66,43)(37,49,56,59)(39,63,68,44)4130 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(15104, 526);