Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C97 ⋊ #2135 |
230 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 3104 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(98,101,104,102)(99,106,108,115)(100,110,113,111)(103,114,118,116)(105,109,121,122)(107,123,125,124)(112,127,117,119)(120,126,129,128);(2,4)(3,6)(5,9)(7,13)(8,15)(10,18)(11,12)(14,22)(16,26)(17,28)(19,31)(20,25)(21,24)(23,35)(27,39)(29,43)(30,45)(32,48)(33,42)(34,41)(36,52)(37,38)(40,56)(44,60)(46,64)(47,66)(49,69)(50,63)(51,62)(53,73)(54,59)(55,58)(57,77)(61,74)(65,70)(67,84)(68,86)(71,83)(72,82)(75,81)(76,80)(78,79)(85,91)(87,93)(88,92)(89,90)(94,97)(95,96)(98,100,104,113)(99,105,108,121)(101,111,102,110)(103,117,118,112)(106,122,115,109)(107,120,125,129)(114,127,116,119)(123,128,124,126);(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)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(98,99,103,107,104,108,118,125)(100,109,112,126,113,122,117,128)(101,106,114,123,102,115,116,124)(105,119,120,110,121,127,129,111)Polycyclic presentation:
2^7,97|a^2=e,b^2=g,c^2=g,d^2=g,e^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^22,c^b=cg,d^b=dg,e^b=efg,h^b=h^96,e^d=eg,h^e=h^96503 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20814);