Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#13100 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C83 | #1898 |
| commutator | C22×C4 | C22×C4×C83 |
| Frattini, cosocle | C2×#41 | C23×C83 |
| prime radical | C2×H(5,𝔽2)×C83 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C83 ◁ C2×C4×C83 ◁ C2×H(5,𝔽2)×C83 ◁ G | |
672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 115 |
| transitive degree | μt(G) = 2656 |
A minimal faithful permutation representation — 3 generators on 115 points:
(1,2,6,9,7,12,5,11)(3,13,17,29,18,31,16,30)(4,19,14,27,21,10,22,24)(8,23,25,20,26,32,28,15)(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);(1,3,14,23)(2,8,24,29)(4,20,6,16)(5,17,21,15)(7,18,22,32)(9,25,10,13)(11,28,19,31)(12,26,27,30);(2,9)(4,21)(5,6)(8,25)(10,24)(11,12)(13,29)(15,20)(16,17)(19,27)(26,28)(30,31)2822 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 6088);