Finite Group DatabasebetaFinite Group Databasebeta| order | 21248 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C83 · Schur multiplier (H2) = C2 |
| direct product | C83×#7438 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C83 | #353 |
| commutator | C2×M16 | C2×C4×C83 |
| Frattini, cosocle | #408 | C22×C83 |
| socle | C2×C83 | #1352 |
| prime radical | C2×#41×C83 | D4 |
| LCS | G ▷ C2×M16 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C83 ◁ C2×C4×C83 ◁ C22×C4×C83 ◁ #408×C83 ◁ G | |
174 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 174 conjugacy classes of subgroups — too many to draw. Only its 42 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:
(1,2,5,10)(3,13,15,39)(4,9,19,32)(6,12,24,37)(7,11,23,38)(8,26,27,47)(14,28,42,35)(16,40,45,62)(17,30,22,51)(18,29,46,55)(20,34,49,36)(21,33,48,60)(25,31,53,59)(41,57,52,64)(43,56,44,61)(50,58,54,63)(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)(4,18,20,47)(5,22,23,52)(6,17)(7,16)(9,30,34,57)(10,35,38,61)(11,29)(12,28)(13,33,40,58)(14,21,43,50)(15,24,45,53)(19,42)(25,41)(26,54)(27,37,55,59)(31,56)(32,39)(36,62)(44,49)(46,48)(51,60)(63,64)2573 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(21248, 426);