Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C83 · Schur multiplier (H2) = C23 |
| direct product | C83×#12877 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C83 | #418 |
| commutator | C2×C8 | C22×C4×C83 |
| Frattini, cosocle | C22×C8 | C23×C83 |
| socle | C2×C83 | #1824 |
| prime radical | C2×#41×C83 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C83 ◁ C22×C4×C83 ◁ C2×#131×C83 ◁ G | |
366 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 366 conjugacy classes of subgroups — too many to draw. Only its 138 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 — 3 generators on 147 points:
(1,2)(3,13,16,8)(4,18)(5,22)(6,28)(7,10)(9,20)(11,26)(12,25)(14,41,42,60)(15,45,23,55)(17,32,47,40)(19,50)(21,53)(24,38)(27,54)(29,37)(30,43,39,52)(31,48,36,57)(33,58)(34,51)(35,59)(44,62,49,64)(46,61,56,63)(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)(2,8)(4,14)(5,23)(6,17)(7,16)(9,30)(10,13)(11,36)(12,32)(15,27)(18,41)(19,42)(20,52)(21,44)(22,45)(24,56)(25,47)(26,57)(28,40)(29,49)(31,35)(33,61)(34,39)(37,64)(38,63)(43,51)(46,58)(48,59)(50,60)(53,62)(54,55);(1,4,6,29,7,19,25,21)(2,9,28,33,10,34,12,38)(3,14,17,49,16,42,47,44)(5,24,26,20,27,58,59,51)(8,30,40,61,13,39,32,63)(11,37,54,50,35,53,22,18)(15,46,48,43,23,56,57,52)(31,62,45,41,36,64,55,60)3818 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 5865);