Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | C97 ⋊ #2108 |
184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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:
(1,2,7,18,36,64,70,91,86,61,75,37,65,87,27,53,77,31,56,78,55,29,5,13,32,23,48,76,44,68,67,14,25,10,21,46,72,92,97,73,93,89,40,58,16,35,42,8,19,38,9,20,43,15,34,41,66,90,81,94,96,59,60,39,3,11,26,50,79,45,69,74,24,49,28,4,12,30,54,83,51,57,82,52,80,84,22,47,62,85,95,71,88,63,33,17,6)(100,108)(104,114)(109,120)(110,121)(115,124)(116,125)(122,128)(126,129);(2,4,6,5)(3,8,10,9)(7,15,17,16)(11,23,25,24)(12,28,29,13)(14,33,26,18)(19,37,38,22)(20,40,42,41)(21,44,39,45)(27,51,52,31)(30,43,55,35)(32,34,49,58)(36,61,63,62)(46,70,60,71)(47,48,75,74)(50,54,67,78)(53,81,82,73)(56,85,83,86)(57,59,77,72)(64,80,88,87)(65,76,84,69)(66,68,89,79)(90,91,93,95)(92,94,96,97)(98,100)(99,104)(101,109)(102,110)(103,108)(105,115)(106,116)(107,114)(111,122)(112,120)(113,121)(117,126)(118,124)(119,125)(123,128)(127,129);(2,3,5,9,6,10,4,8)(7,14,16,18,17,26,15,33)(11,22,24,38,25,37,23,19)(12,27,13,31,29,52,28,51)(20,39,41,44,42,21,40,45)(30,54,35,50,55,78,43,67)(32,57,58,72,49,77,34,59)(36,60,62,70,63,46,61,71)(47,73,74,82,75,81,48,53)(56,84,86,76,83,65,85,69)(64,68,87,66,88,79,80,89)(90,94,95,92,93,97,91,96)(98,99,101,105,102,106,111,117,103,107,112,118,113,119,123,127)(100,104,109,115,110,116,122,126,108,114,120,124,121,125,128,129)Polycyclic presentation:
2^7,97|a^2=d,b^2=e,d^2=f,f^2=g,h^a=h^64,c^b=cg,h^b=h^75,h^d=h^22,h^e=h^96,h^f=h^96200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21673);