Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C79 · Schur multiplier (H2) = C25 |
| direct product | C2×C79×#2151 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8×C79 | C23 |
| commutator | C22 | C23×C8×C79 |
| Frattini, cosocle | C2×C8 | C24×C79 |
| socle | C23×C79 | C22×C8 |
| prime radical | C24×C79 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8×C79 ◁ G | |
734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 117 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 4 generators on 117 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79)(80,81);(83,85)(86,87,89,93,90,94,99,105,91,95,100,106,101,107,111,115)(88,92,97,103,98,104,110,114,96,102,108,112,109,113,116,117);(82,83,84,85)(86,88)(87,92)(89,97)(90,98)(91,96)(93,103)(94,104)(95,102)(99,110)(100,108)(101,109)(105,114)(106,112)(107,113)(111,116)(115,117);(88,96)(92,102)(97,108)(98,109)(103,112)(104,113)(110,116)(114,117)8848 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 26384);