Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22⋊C16×C97 ⋊ C2 |
| Sylow | C97 ⋊ #1315 |
116 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 37 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) = 6208 |
A minimal faithful permutation representation — 2 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)(98,100)(99,102)(101,105)(103,104)(106,108)(107,113)(109,122)(110,124)(111,119)(112,121)(114,130)(115,132)(116,118)(117,129)(120,126)(123,136)(125,127)(128,134)(131,135)(133,137);(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)(98,99)(100,104,105,102)(101,103)(106,107,110,115,111,116,125,133,112,117,126,134,109,114,123,131)(108,118,120,135,119,129,136,132,121,130,124,137,122,113,127,128)Polycyclic presentation:
2^7,97|a^2=d,c^2=g,d^2=f,f^2=g,b^a=bc,c^a=ce,c^b=cg,d^b=deg,h^b=h^961958 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 2435);