Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7337 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C83 | #99 |
| commutator | C2×C4 | C2×C16×C83 |
| Frattini, cosocle | C2×M32 | C22×C83 |
| socle | C2×C83 | #1314 |
| prime radical | C2×#41×C83 | C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C83 ◁ C2×C8×C83 ◁ C2×M32×C83 ◁ G | |
154 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 154 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 147 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 2 generators on 147 points:
(1,2,5,11,6,13,25,38,7,10,24,34,27,35,51,58,8,12,26,37,28,39,52,60,4,19,21,36,22,46,48,59)(3,14,16,40,17,41,44,61,15,9,29,30,42,32,53,55,18,33,23,49,45,57,50,63,20,31,43,54,47,56,62,64)(65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147);(1,3)(2,9)(4,20)(5,23)(6,17)(7,15)(8,18)(10,33)(11,30)(12,31)(13,32)(14,19)(16,26)(21,29)(22,47)(24,43)(25,50)(27,42)(28,45)(34,49)(35,57)(36,40)(37,54)(38,55)(39,56)(41,46)(44,52)(48,53)(51,62)(58,63)(59,61)(60,64)4316 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 325);