Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| isoclinism | isoclinic class of D79 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C32 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×C32×Dic79 |
| derived, Sylow | C79 ⋊ C2×C4×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C32 | D79 |
| commutator, hypocenter | C79 | C2×C4×C32 |
| Frattini | C2×C16 | C22×D79 |
| Fitting | C22×C32×C79 | C2 |
| socle, prime radical | C23×C79 | C2×C16 |
| cosocle | C2×C16×C79 | C23 |
| LCS | G ▷ C79 | |
| UCS | C1 ◁ C22×C32 | |
270 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 20224 |
A minimal faithful permutation representation — 3 generators on 117 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(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)(82,83,84,85);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(86,87,88,92,89,93,96,102,90,94,97,103,99,105,108,112,91,95,98,104,100,106,109,113,101,107,110,114,111,115,116,117);(80,81)5248 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 89737);