Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 6 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | #1415 |
| commutator | C32 | C2×C4×C83 |
| Frattini, cosocle | C2×C32 | C22×C83 |
| prime radical | C22×C83 | SD64 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C2×C8×C83 ◁ C2×C16×C83 ◁ C2×C32×C83 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 211 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 2 generators on 211 points:
(1,2,5,11,8,15,32,14)(3,16,18,62,21,66,29,65)(4,22,25,78,28,82,90,81)(6,33,30,98,36,48,100,101)(7,37,31,13,39,54,76,60)(9,40,41,17,45,68,44,71)(10,46,49,92,52,111,51,26)(12,55,53,24,58,87,57,35)(19,73,72,118,74,114,96,124)(20,75,38,64,77,113,97,122)(23,56,83,121,69,61,88,120)(27,93,89,80,95,126,125,116)(34,63,103,115,67,47,99,112)(42,94,105,102,108,128,107,85)(43,70,106,104,109,123,59,84)(50,86,117,127,119,91,79,110)(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);(1,3,8,21)(2,9,15,45)(4,23,28,69)(5,29,32,18)(6,34,36,67)(7,38,39,97)(10,47,52,63)(11,44,14,41)(12,56,58,61)(13,59,60,106)(16,22,66,82)(17,26,71,92)(19,24,74,35)(20,76,77,31)(25,88,90,83)(27,94,95,128)(30,99,100,103)(33,42,48,108)(37,43,54,109)(40,46,68,111)(49,115,51,112)(50,118,119,124)(53,120,57,121)(55,96,87,72)(62,81,65,78)(64,93,122,126)(70,110,123,127)(73,79,114,117)(75,80,113,116)(84,86,104,91)(85,125,102,89)(98,107,101,105)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 530);