Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C83 · Schur multiplier (H2) = C24 |
| direct product | C83×#24707 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C83 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24×C83 |
| prime radical | C24×C83 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C83 ◁ C22×C42×C83 ◁ G | |
784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 111 |
| transitive degree | μt(G) = 10624 |
A minimal faithful permutation representation — 4 generators on 111 points:
(5,6,8,9)(7,11,12,10)(13,14,16,20,17,19,25,27)(15,21,22,26,23,28,24,18)(29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111);(1,2)(3,4)(6,9)(10,11);(2,4)(13,15,17,23)(14,18,19,26)(16,24,25,22)(20,28,27,21);(5,7,8,12)(6,10,9,11)(14,19)(18,26)(20,27)(21,28)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 17695);