Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C2×Q16×C97 ⋊ C4 |
| Sylow | C97 ⋊ #2141 |
220 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 220 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(1,2,5,10,19,32,49,70,64,45,62,80,72,47,67,85,92,83,63,42,25,12,3,7,14,23,36,53,74,60,43,28,41,58,78,68,87,94,96,90,81,59,38,21,8,16,27,40,57,77,56,39,26,15,24,37,54,75,89,95,97,93,86,79,55,34,17,29,44,61,73,52,35,22,13,6,11,20,33,50,71,88,91,84,66,82,76,51,30,46,65,69,48,31,18,9,4)(98,101,104,103)(99,106,108,119)(100,110,113,112)(102,114,118,117)(105,120,121,122)(107,115,124,125)(109,123,126,129)(111,127,128,116);(98,100,104,113)(99,105,108,121)(101,112,103,110)(102,116,118,127)(106,122,119,120)(107,123,124,129)(109,125,126,115)(111,114,128,117);(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,28)(14,21,22,24)(16,25,26,20)(19,30,31,45)(23,34,35,41)(27,38,39,37)(29,42,43,33)(32,47,48,66)(36,51,52,62)(40,55,56,58)(44,59,60,54)(46,63,64,50)(49,68,69,86)(53,72,73,82)(57,76,77,80)(61,79,74,78)(65,81,70,75)(67,83,84,71)(85,90,91,89)(87,92,93,88)(94,96,97,95)(98,99)(100,109)(101,106)(102,115)(103,119)(104,108)(105,111)(107,117)(110,123)(112,129)(113,126)(114,124)(116,122)(118,125)(120,127)(121,128)Polycyclic presentation:
2^7,97|a^2=e,b^2=g,c^2=g,d^2=fg,f^2=g,b^a=bd,d^a=df,f^a=fg,h^a=h^22,c^b=cg,d^b=df,f^b=fg,h^e=h^96368 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 20820);