Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C83 · Schur multiplier (H2) = C27×C4 |
| direct product | C83×#60828 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C83 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25×C83 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C83 ◁ C4×#41×C83 ◁ G | |
2018 conjugacy classes of subgroups.
| minimal degree | μ(G) = 123 |
| transitive degree | μt(G) = 5312 |
A minimal faithful permutation representation — 5 generators on 123 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,19)(12,23)(13,15)(14,27)(16,22)(17,24)(18,26)(20,34)(21,32)(25,31)(28,29)(30,40)(33,35)(36,37)(38,39)(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);(1,3,4,8)(2,5,6,7)(10,15)(16,29)(17,23)(18,27)(19,32)(30,37)(31,38)(33,34);(3,8)(5,7)(9,11,13,21)(10,16,15,29)(12,20,24,35)(14,28,26,22)(17,30,23,37)(18,32,27,19)(25,40,39,36)(31,34,38,33);(2,6)(5,7)(9,12)(10,17)(11,20)(13,24)(14,25)(15,23)(16,30)(18,31)(19,33)(21,35)(22,36)(26,39)(27,38)(28,40)(29,37)(32,34);(12,24)(17,23)(20,35)(25,39)(30,37)(31,38)(33,34)(36,40)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 53816);