Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C97 ⋊ M128 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C16 | C2×#87604 |
| commutator, socle | C2×C97 | C2×C32 |
| Fitting | C32×C97 | C4 |
| prime radical | C2×D97 | C32 |
| cosocle | #15690716 | C22 |
| hypercenter | C32 | #87604 |
| hypocenter | C97 | M128 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C16 ◁ C32 | |
38 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 2 generators on 161 points:
(1,2)(3,11)(4,5)(6,7)(8,24)(9,10)(12,20)(13,14)(15,16)(17,41)(18,19)(21,37)(22,23)(25,33)(26,27)(28,29)(30,62)(31,32)(34,58)(35,36)(38,54)(39,40)(42,50)(43,44)(45,46)(47,82)(48,49)(51,80)(52,53)(55,78)(56,57)(59,75)(60,61)(63,71)(64,65)(66,67)(68,79)(69,70)(72,76)(73,74)(81,89)(83,88)(84,85)(86,87)(90,95)(91,92)(93,94)(96,97)(99,105)(106,120)(107,121)(108,122)(109,123)(124,140)(125,141)(126,142)(127,143)(128,144)(129,145)(146,155)(147,156)(148,157)(149,158)(159,161);(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)(98,99,100,106,101,107,110,124,102,108,111,125,114,127,130,146,103,109,112,126,115,128,131,147,117,129,133,148,136,149,150,159,104,105,113,120,116,121,132,140,118,122,134,141,137,143,151,155,119,123,135,142,138,144,152,156,139,145,153,157,154,158,160,161)Polycyclic presentation:
2^7,97|a^2=c,c^2=d,d^2=e,e^2=f,f^2=g,b^a=bg,h^a=h^22,h^b=h^96,h^c=h^96848 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 19680);