Finite Group DatabasebetaFinite Group Databasebeta| order | 3616 |
| automorphism | Aut(G) order 1619968 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| dihedral | D1808 |
| Fitting | C16×C113 ⋊ C2 |
| Sylow | C113 ⋊ D16 |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 129 |
| transitive degree | μt(G) = 1808 |
A minimal faithful permutation representation — 2 generators on 129 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,129)(19,128)(20,127)(21,126)(22,125)(23,124)(24,123)(25,122)(26,121)(27,120)(28,119)(29,118)(30,117)(31,116)(32,115)(33,114)(34,113)(35,112)(36,111)(37,110)(38,109)(39,108)(40,107)(41,106)(42,105)(43,104)(44,103)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)Polycyclic presentation:
2^5,113|c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^112,c^b=cd,d^b=de,f^b=f^112907 conjugacy classes — view the full character table.
| maximal subgroup of | D5424, D9040, D12656, … |
| maximal quotient of | D5424, D9040, D12656, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(3616, 88);