Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | #568×C97 ⋊ C2 |
| Sylow | C97 ⋊ #2889 |
704 conjugacy classes of subgroups; only the 249 normal subgroups are shown.
This group has 704 conjugacy classes of subgroups — too many to draw. Only its 249 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 — 4 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)(101,112)(105,118)(109,123)(113,125)(116,122)(119,128)(124,126)(127,129);(98,101)(99,105)(100,109)(102,113)(103,112)(104,116)(106,118)(107,119)(108,122)(110,124)(111,123)(114,126)(115,125)(117,127)(120,128)(121,129);(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)(98,100,103,111)(99,104,106,108)(101,109,112,123)(102,114,115,110)(105,116,118,122)(107,121,120,117)(113,126,125,124)(119,129,128,127);(98,99,102,107,103,106,115,120)(100,108,110,121,111,104,114,117)(101,105,113,119,112,118,125,128)(109,122,124,129,123,116,126,127)Polycyclic presentation:
2^7,97|a^2=f,b^2=g,f^2=g,b^a=be,e^a=eg,f^b=fg,h^b=h^96,d^c=dg1673 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 12992);