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 | C2 | C2×#15690911 |
| commutator, socle | C2×C97 | C2×C32 |
| Fitting, prime radical | C22×C97 | C32 |
| cosocle | #15690909 | C22 |
| hypercenter | C22 | #15690911 |
| hypocenter | C97 | M128 |
| LCS | G ▷ C2×C97 ▷ C97 | |
| UCS | C1 ◁ C2 ◁ C22 | |
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:
(1,2,8,28,49,94,92,83,4,17,51,95,42,40,86,89,24,59,43,74,73,60,6,22,61,91,88,14,3,13,9,25,54,19,11,36,38,82,39,85,87,64,32,76,26,70,10,34,79,90,18,33,77,69,66,75,23,65,46,84,62,67,57,35,5,20,55,31,41,12,37,80,47,81,52,21,56,29,30,15,44,68,97,96,71,78,53,58,50,16,45,48,63,93,72,27,7)(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);(2,3,4,9,5,10,14,29,6,11,15,30,18,32,42,70,7,12,16,31,19,33,37,73,21,35,43,74,34,75,78,77)(8,23,24,62,25,63,66,65,26,41,67,84,28,71,61,45,27,64,68,82,55,94,76,87,69,13,38,39,72,40,81,83)(17,46,47,56,48,88,90,86,49,89,54,22,57,58,95,59,50,85,91,60,92,80,79,96,93,97,20,52,36,51,53,44)(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^28,h^c=h^8,h^d=h^64,h^e=h^22,h^f=h^9692 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12416, 21815);