Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C4 |
No aliases for this group.
| Sylow | C97 ⋊ #2245 |
132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 137 |
| transitive degree | μt(G) = 12416 |
A minimal faithful permutation representation — 3 generators on 137 points:
(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)(99,103)(102,105)(106,109,110,124,126,136,120,123,113,125,127,122,112,121,111,108)(107,115,116,133,128,137,129,132,119,134,135,131,118,130,117,114);(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)(98,100,101,104)(99,102,103,105)(106,108,120,136,112,122,110,109,113,123,111,121,126,124,127,125)(107,114,129,137,118,131,116,115,119,132,117,130,128,133,135,134);(98,99)(100,102)(101,103)(104,105)(106,107,113,119)(108,115,123,134)(109,114,125,132)(110,117,127,129)(111,116,120,135)(112,128,126,118)(121,133,136,131)(122,137,124,130)Polycyclic presentation:
2^7,97|a^2=g,b^2=eg,c^2=d,d^2=fg,e^2=f,f^2=g,b^a=bd,c^a=ce,d^a=df,e^a=efg,f^a=fg,c^b=cg,h^c=h^961082 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 6754);