Finite Group DatabasebetaFinite Group Databasebeta| order | 12928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | SD32×C101 ⋊ C4 |
| Sylow | C101 ⋊ #2180 |
142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 3232 |
A minimal faithful permutation representation — 3 generators on 133 points:
(1,2,8,4)(3,6,18,14)(5,7,16,10)(9,12,32,24)(11,17,28,20)(13,15,30,26)(19,22,50,38)(21,31,42,34)(23,25,48,44)(27,29,46,40)(33,36,69,56)(35,49,53,52)(37,39,68,60)(41,47,64,58)(43,45,66,62)(51,54,67,71)(55,57,83,74)(59,61,82,78)(63,65,80,76)(70,72,81,85)(73,75,93,88)(77,79,92,90)(84,86,91,95)(87,89,99,97)(94,96,98,101)(102,105,107,106,108,119,124,121)(103,110,112,118,113,128,122,111)(104,115,123,120,116,132,117,133)(109,125,130,131,126,114,127,129);(102,104)(103,109)(105,115)(106,120)(107,123)(108,116)(110,125)(111,129)(112,130)(113,126)(114,128)(117,124)(118,131)(119,132)(121,133)(122,127);(2,3,5,4)(6,9,11,10)(7,13,14,8)(12,19,21,20)(15,23,24,18)(16,17,27,26)(22,33,35,34)(25,37,38,32)(28,31,41,40)(29,43,44,30)(36,51,53,52)(39,55,56,50)(42,49,54,58)(45,59,60,48)(46,47,63,62)(57,70,71,69)(61,73,74,68)(64,67,72,76)(65,77,78,66)(75,84,85,83)(79,87,88,82)(80,81,86,90)(89,94,95,93)(91,96,97,92)(98,100,101,99)(102,103)(104,114,116,125)(105,118)(106,110)(107,122)(108,113)(109,115,126,132)(111,119)(112,124)(117,129,123,131)(120,130,133,127)(121,128)Polycyclic presentation:
2^7,101|a^2=e,c^2=ef,d^2=f,f^2=g,b^a=bd,c^a=cf,d^a=df,f^a=fg,h^a=h^10,d^b=dfg,e^b=eg,f^b=fg,d^c=dg,h^c=h^91,h^e=h^100298 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12928, 20916);