Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C97 ⋊ #2047 |
256 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 133 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 133 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)(99,101)(102,105,106,118,107,119,122,133)(103,109,110,126,111,116,120,131)(104,113,114,130,115,117,121,132)(108,123,112,128,124,125,127,129);(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)(102,104,107,115)(103,108,111,124)(105,117,119,113)(106,121,122,114)(109,125,116,123)(110,127,120,112)(118,130,133,132)(126,128,131,129);(98,99)(100,101)(102,103,107,111)(104,112,115,127)(105,116,119,109)(106,120,122,110)(108,121,124,114)(113,129,117,128)(118,126,133,131)(123,130,125,132)Polycyclic presentation:
2^7,97|a^2=g,b^2=g,c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,c^b=cf,d^b=dg,h^b=h^96,e^c=eg,f^c=fg1082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 5776);