Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C64 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | D101⋊C64 |
| derived, Sylow | C101 ⋊ C2×C64 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | #87938 |
| commutator, hypocenter | C101 | C2×C64 |
| Frattini | C16 | C2×#87938 |
| Fitting | C32×C101 | C4 |
| socle | C2×C101 | C2×C32 |
| prime radical | C2×D101 | C32 |
| cosocle | #15692161 | C22 |
| LCS | G ▷ C101 | |
| UCS | C1 ◁ C32 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 165 |
| transitive degree | μt(G) = 6464 |
A minimal faithful permutation representation — 2 generators on 165 points:
(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,80)(48,49)(51,77)(52,53)(55,73)(56,57)(59,69)(60,61)(63,64)(66,91)(67,68)(70,88)(71,72)(74,84)(75,76)(78,79)(81,98)(82,83)(85,95)(86,87)(89,90)(92,101)(93,94)(96,97)(99,100);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,65)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,49,63,50)(53,66,67,80)(57,70,71,77)(61,74,75,73)(64,68,78,69)(72,81,82,91)(76,85,86,88)(79,83,89,84)(87,92,93,98)(90,94,96,95)(97,99,100,101)(102,103,104,109,105,110,114,124,106,111,115,125,118,128,134,144,107,112,116,126,119,129,135,145,121,131,137,147,140,150,154,159,108,113,117,127,120,130,136,146,122,132,138,148,141,151,155,160,123,133,139,149,142,152,156,161,143,153,157,162,158,163,164,165)Polycyclic presentation:
2^7,101|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,h^a=h^10,h^b=h^100,h^c=h^100928 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 19678);