Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #1537 |
194 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 6208 |
A minimal faithful permutation representation — 3 generators on 129 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)(47,82)(48,49)(51,80)(52,53)(55,78)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,79)(69,70)(72,76)(73,74)(81,89)(83,88)(84,85)(86,87)(90,95)(91,92)(93,94)(96,97)(114,117,119,118)(115,121,122,126)(116,124,125,127)(120,128,129,123);(98,100,102,108)(99,103,104,106)(101,109,110,107)(105,113,112,111)(114,116)(115,120)(117,124)(118,127)(119,125)(121,128)(122,129)(123,126);(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)(98,99,101,105,102,104,110,112)(100,106,107,113,108,103,109,111)(114,115)(116,123,125,128)(117,126)(118,121)(119,122)(120,124,129,127)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=g,d^2=g,e^2=f,b^a=bd,c^a=cg,d^a=df,h^a=h^22,d^b=dg,e^b=efg,h^c=h^96,h^e=h^96362 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20002);