Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| Fitting | #548×C97 ⋊ C2 |
| Sylow | C97 ⋊ #3524 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D97 |
| commutator, socle, cosocle | C22×C97 | C25 |
| Fitting | #548×C97 | C2 |
| hypercenter | #548 | D97 |
| hypocenter | C97 | #3524 |
| LCS | G ▷ C22×C97 ▷ C97 | |
| UCS | C1 ◁ C22 ◁ #548 | |
1030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 121 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 5 generators on 121 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)(98,100,101,104)(99,103,102,105)(106,110,111,109)(107,113,114,119)(108,117,118,116)(112,120,115,121);(106,109,111,110)(107,113,114,119)(108,117,118,116)(112,121,115,120);(99,102)(103,105)(106,109,111,110)(107,113,114,119)(108,116,118,117)(112,120,115,121);(106,108,111,118)(107,112,114,115)(109,116,110,117)(113,120,119,121);(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)(100,103)(101,102)(104,105)(106,107)(108,115)(109,113)(110,119)(111,114)(112,118)(116,121)(117,120)Polycyclic presentation:
2^7,97|b^2=f,c^2=f,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,h^a=h^96,d^b=df,h^e=h^961244 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 18620);