Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7514 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C83 | D4 |
| commutator | C4 | C2×C32×C83 |
| Frattini, cosocle | C2×C32 | C22×C83 |
| socle | C2×C83 | C22⋊C32 |
| prime radical | #41×C83 | C16 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C32×C83 ◁ C2×C32×C83 ◁ G | |
88 conjugacy classes of subgroups.
| minimal degree | μ(G) = 211 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 2 generators on 211 points:
(1,2,5,12,6,13,28,47,7,14,29,48,32,51,75,96,8,15,30,49,33,52,76,97,35,54,78,99,80,101,116,124,9,16,31,50,34,53,77,98,36,55,79,100,81,102,117,125,4,11,24,43,25,44,69,91,26,45,70,92,71,93,112,122)(3,17,19,56,20,57,62,103,21,58,63,104,64,105,109,126,18,41,59,46,60,87,106,94,61,88,107,95,108,120,127,123,22,42,27,72,65,89,73,113,66,90,74,114,110,121,115,128,23,10,37,38,67,39,82,84,68,40,83,85,111,86,118,119)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211);(1,3)(2,10)(4,23)(5,27)(6,20)(7,21)(8,18)(9,22)(11,42)(12,46)(13,39)(14,40)(15,17)(16,41)(19,31)(24,59)(25,67)(26,68)(28,73)(29,74)(30,37)(32,64)(33,60)(34,65)(35,61)(36,66)(38,50)(43,56)(44,89)(45,90)(47,94)(48,95)(49,72)(51,86)(52,57)(53,87)(54,58)(55,88)(62,77)(63,79)(69,106)(70,107)(71,111)(75,115)(76,82)(78,83)(80,108)(81,110)(84,98)(85,100)(91,103)(92,104)(93,121)(96,123)(97,113)(99,114)(101,105)(102,120)(109,117)(112,127)(116,118)(119,125)(122,126)(124,128)9296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 502);