Finite Group DatabasebetaFinite Group Databasebeta| order | 20224 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of D64 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C79 · Schur multiplier (H2) = C22 |
| direct product | C79×#13741 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C79 | D32 |
| commutator, Frattini, cosocle | C32 | C23×C79 |
| socle | C2×C79 | C2×D32 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C79 ◁ C2×C4×C79 ◁ C2×C8×C79 ◁ C2×C16×C79 ◁ C2×C32×C79 ◁ G | |
128 conjugacy classes of subgroups.
| minimal degree | μ(G) = 207 |
| transitive degree | μt(G) = 10112 |
A minimal faithful permutation representation — 3 generators on 207 points:
(1,2)(3,16)(4,10)(5,27)(6,33)(7,36)(8,15)(9,18)(11,30)(12,29)(13,25)(14,26)(17,56)(19,65)(20,68)(21,70)(22,58)(23,71)(24,77)(28,50)(31,92)(32,84)(34,46)(35,96)(37,57)(38,45)(39,59)(40,86)(41,85)(42,64)(43,74)(44,73)(47,106)(48,109)(49,91)(51,113)(52,90)(53,62)(54,80)(55,63)(60,107)(61,116)(66,120)(67,119)(69,121)(72,104)(75,82)(76,83)(78,102)(79,95)(81,89)(87,114)(88,112)(93,111)(94,122)(97,110)(98,105)(99,123)(100,126)(101,117)(103,127)(108,128)(115,118)(124,125)(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);(1,3)(2,9)(4,17)(5,28)(6,34)(7,37)(8,22)(10,39)(11,46)(12,50)(13,53)(14,55)(15,42)(16,27)(18,30)(19,29)(20,25)(21,26)(23,72)(24,78)(31,93)(32,88)(33,40)(35,98)(36,41)(38,61)(43,102)(44,104)(45,101)(47,107)(48,110)(49,105)(51,114)(52,112)(54,115)(56,71)(57,92)(58,84)(59,74)(60,73)(62,106)(63,109)(64,91)(65,81)(66,113)(67,90)(68,82)(69,80)(70,83)(75,86)(76,87)(77,99)(79,97)(85,89)(94,123)(95,111)(96,100)(103,119)(108,120)(116,122)(117,127)(118,128)(121,125)(124,126);(1,4,8,26)(2,10,15,14)(3,17,22,21)(5,23,32,76)(6,24,35,79)(7,25,38,80)(9,39,42,55)(11,43,49,48)(12,44,52,51)(13,45,54,36)(16,56,58,70)(18,59,64,63)(19,60,67,66)(20,61,69,37)(27,71,84,83)(28,72,88,87)(29,73,90,113)(30,74,91,109)(31,75,94,124)(33,77,96,95)(34,78,98,97)(40,99,100,111)(41,53,101,115)(46,102,105,110)(47,103,108,81)(50,104,112,114)(57,68,116,121)(62,117,118,85)(65,107,119,120)(82,122,125,92)(86,123,126,93)(89,106,127,128)5530 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(20224, 6729);