Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Fitting | #100×C97 ⋊ C4 |
| Sylow | C97 ⋊ #1757 |
284 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 284 conjugacy classes of subgroups — too many to draw. Only its 86 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:
(1,2,7,3)(4,6,17,13)(5,8,15,9)(10,12,31,27)(11,18,25,19)(14,16,29,23)(20,22,49,41)(21,32,39,33)(24,30,45,37)(26,28,47,43)(34,36,71,59)(35,50,57,51)(38,48,63,55)(40,42,69,65)(44,46,67,61)(52,54,89,81)(53,72,79,73)(56,70,84,77)(58,60,88,85)(62,68,78,83)(64,66,75,87)(74,76,86,91)(80,82,95,93)(90,92,94,97)(98,101,102,112,103,113,114,125)(99,105,106,118,107,119,120,128)(100,109,111,124,110,123,115,126)(104,116,117,127,108,121,122,129);(2,4,5,3)(6,10,11,9)(7,8,14,13)(12,20,21,19)(15,18,24,23)(16,26,27,17)(22,34,35,33)(25,32,38,37)(28,40,41,31)(29,30,44,43)(36,52,53,51)(39,50,56,55)(42,58,59,49)(45,48,62,61)(46,64,65,47)(54,74,75,73)(57,72,78,77)(60,80,81,71)(63,70,84,83)(66,76,85,69)(67,68,79,87)(82,90,91,89)(86,92,93,88)(94,96,97,95)(98,100)(99,104,107,108)(101,109)(102,111)(103,110)(105,116,119,121)(106,117,120,122)(112,124)(113,123)(114,115)(118,127,128,129)(125,126);(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)(42,49,59,58)(45,61,62,48)(46,47,65,64)(54,73,75,74)(57,77,78,72)(60,71,81,80)(63,83,84,70)(66,69,85,76)(67,87,79,68)(82,89,91,90)(86,88,93,92)(94,95,97,96)(98,99,102,106,103,107,114,120)(100,108,111,122,110,104,115,117)(101,105,112,118,113,119,125,128)(109,121,124,129,123,116,126,127)Polycyclic presentation:
2^7,97|a^2=e,b^2=f,c^2=e,e^2=g,b^a=bd,f^a=fg,h^a=h^22,d^b=dg,h^b=h^75,h^c=h^22,h^e=h^96,h^f=h^96308 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20267);