Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8×C83 · Schur multiplier (H2) = C22 |
| direct product | C83×#7172 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C83 | D8 |
| commutator | C8 | C4×C8×C83 |
| Frattini, cosocle | C2×C4×C8 | C22×C83 |
| socle | C22×C83 | #340 |
| prime radical | C23×C83 | #103 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C83 ◁ C2×C42×C83 ◁ C2×C4×C8×C83 ◁ G | |
246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 2 generators on 123 points:
(1,2,4,7,5,8,3,6)(9,10,13,18,15,21,27,20)(11,22,24,36,26,37,38,31)(12,19,14,17,28,34,29,35)(16,25,30,39,33,40,32,23)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(2,6,8,7)(9,11,14,25,13,24,28,39,15,26,29,40,27,38,12,23)(10,16,19,31,20,32,35,37,21,33,34,36,18,30,17,22)7304 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 160);