Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C43×C83 · Schur multiplier (H2) = C42 |
| direct product | C83×#12542 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C83 | C42 |
| commutator | C4 | C43×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| socle | C2×C83 | #1726 |
| prime radical | C23×C83 | #131 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C16×C83 ◁ G | |
350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 3 generators on 147 points:
(1,2,5,10,6,11,21,32,7,12,22,33,23,34,45,56)(3,8,14,25,16,27,38,49,15,26,37,48,39,50,58,62)(4,9,18,29,20,31,43,54,19,30,42,53,44,55,60,64)(13,24,35,46,17,28,40,51,36,47,57,61,41,52,59,63)(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);(1,3)(2,8)(4,17,19,41)(5,14)(6,16)(7,15)(9,28,30,52)(10,25)(11,27)(12,26)(13,20,36,44)(18,40,42,59)(21,38)(22,37)(23,39)(24,31,47,55)(29,51,53,63)(32,49)(33,48)(34,50)(35,43,57,60)(45,58)(46,54,61,64)(56,62);(1,4)(2,9)(3,13,15,36)(5,18)(6,20)(7,19)(8,24,26,47)(10,29)(11,31)(12,30)(14,35,37,57)(16,17,39,41)(21,43)(22,42)(23,44)(25,46,48,61)(27,28,50,52)(32,54)(33,53)(34,55)(38,40,58,59)(45,60)(49,51,62,63)(56,64)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5530);