Finite Group DatabasebetaFinite Group Databasebeta| order | 12416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C97 ⋊ #2258 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×#11715906 |
| commutator, socle | C2×C97 | C22×C16 |
| Fitting, prime radical | #41×C97 | C8 |
| cosocle | #13095514 | C23 |
| hypercenter | #41 | #11715906 |
| hypocenter | C97 | #2258 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C4 ◁ #41 | |
106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 161 |
| transitive degree | μt(G) = 6208 |
A minimal faithful permutation representation — 3 generators on 161 points:
(1,2,6,15,31,58,82,93,78,63,69,41,66,85,46,74,71,24,50,72,54,25,4,11,27,43,70,89,59,84,61,12,21,8,18,39,65,90,96,67,92,87,37,52,13,29,35,16,32,40,7,17,36,28,53,34,60,88,91,95,97,81,55,33,3,9,22,45,73,38,62,68,20,44,26,10,23,47,75,79,49,51,80,48,76,77,19,42,56,83,94,64,86,57,30,14,5)(100,110)(105,120)(111,130)(112,131)(113,132)(121,140)(122,141)(123,142)(133,150)(134,151)(135,152)(143,155)(144,156)(145,157)(153,160)(158,161);(98,100)(99,105)(101,111)(102,112)(103,113)(104,110)(106,121)(107,122)(108,123)(109,120)(114,133)(115,134)(116,130)(117,135)(118,131)(119,132)(124,143)(125,144)(126,140)(127,145)(128,141)(129,142)(136,153)(137,150)(138,151)(139,152)(146,158)(147,155)(148,156)(149,157)(154,160)(159,161);(2,3,4,7,5,8,10,16)(6,12,13,15,14,22,28,30)(9,19,20,40,21,41,43,32)(11,24,25,48,26,49,23,46)(17,33,34,59,35,18,37,38)(27,51,52,65,44,71,53,81)(29,45,54,72,36,61,47,75)(31,55,56,82,57,39,63,64)(42,67,68,80,69,91,70,74)(50,77,78,89,79,66,83,62)(58,84,85,60,86,73,76,87)(88,95,94,90,92,96,93,97)(98,99,101,106,102,107,114,124,103,108,115,125,117,127,136,146,104,109,116,126,118,128,137,147,119,129,138,148,139,149,154,159)(100,105,111,121,112,122,133,143,113,123,134,144,135,145,153,158,110,120,130,140,131,141,150,155,132,142,151,156,152,157,160,161)Polycyclic presentation:
2^7,97|a^2=d,d^2=e,e^2=f,f^2=g,h^a=h^64,c^b=cg,h^d=h^22,h^e=h^96200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21743);