Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #1663 |
172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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) = 12416 |
A minimal faithful permutation representation — 3 generators on 129 points:
(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,80,82,84,86,88,90,92,94,96,97,95,93,91,89,87,85,83,81,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(98,100)(99,103)(101,110)(102,107)(104,113)(105,112)(106,108)(109,111)(114,117,119,115)(116,123,125,120)(118,121,128,126)(122,127,129,124);(98,99,102,105)(100,103,107,112)(101,109,108,104)(106,113,110,111)(114,116,119,125)(115,120,117,123)(118,127,128,124)(121,129,126,122);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(80,81)(82,83)(84,85)(86,87)(88,89)(90,91)(92,93)(94,95)(96,97)(100,106)(101,108)(103,111)(104,109)(107,110)(112,113)(114,115,119,117)(116,122,125,129)(118,126,128,121)(120,127,123,124)Polycyclic presentation:
2^7,97|a^2=f,b^2=g,c^2=f,d^2=f,e^2=fg,b^a=bd,c^a=ce,d^a=df,e^a=efg,h^a=h^96,d^b=df,e^b=eg,d^c=dg,e^c=efg932 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 3838);