Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64×C79 · Schur multiplier (H2) = C1 |
| direct product | C79×M256 |
| subgroup | G/H | |
|---|---|---|
| center | C64×C79 | C22 |
| commutator | C2 | C2×C64×C79 |
| Frattini, cosocle | C64 | C22×C79 |
| socle | C2×C79 | C2×C64 |
| prime radical | C22×C79 | C64 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C64×C79 ◁ G | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 2 generators on 207 points:
(1,2,3,10,4,11,15,35,5,12,16,36,20,39,45,75,6,13,17,37,21,40,46,76,24,42,49,78,55,81,85,110,7,14,18,38,22,41,47,77,25,43,50,79,56,82,86,111,27,44,52,80,58,83,88,112,61,84,91,113,95,114,115,126,8,9,19,30,23,31,48,65,26,32,51,66,57,69,87,100,28,33,53,67,59,70,89,101,62,72,92,103,96,106,116,121,29,34,54,68,60,71,90,102,63,73,93,104,97,107,117,122,64,74,94,105,98,108,118,123,99,109,119,124,120,125,127,128)(129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,172,173,174,175,176,177,178,179,180,181,182,183,184,185,186,187,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207);(2,9)(10,30)(11,31)(12,32)(13,33)(14,34)(35,65)(36,66)(37,67)(38,68)(39,69)(40,70)(41,71)(42,72)(43,73)(44,74)(75,100)(76,101)(77,102)(78,103)(79,104)(80,105)(81,106)(82,107)(83,108)(84,109)(110,121)(111,122)(112,123)(113,124)(114,125)(126,128)12640 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 538);