Finite Group DatabasebetaFinite Group Databasebeta| order | 13516 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C31⋊#89137 |
| Fitting, socle, derived, Hall | C31×C109 ⋊ C4 |
| Sylow | C109 ⋊ Dic31 |
| Sylow | C31 ⋊ #89137 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C31×C109 | C4 |
| prime radical, cosocle | C31×D109 | C2 |
| LCS | G ▷ C31×C109 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 140 |
| transitive degree | μt(G) = 3379 |
A minimal faithful permutation representation — 2 generators on 140 points:
(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)(49,68,69,85)(53,72,73,86)(57,76,77,82)(61,70,80,79)(65,83,84,75)(67,87,88,71)(74,89,90,98)(78,92,93,99)(81,91,95,97)(94,100,101,105)(96,102,103,106)(104,107,108,109)(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140);(1,2,5,10,19,32,49,70,55,34,17,29,44,61,81,96,104,107,100,89,68,84,64,45,62,82,97,90,73,52,35,22,13,6,11,20,33,50,71,83,59,38,21,8,16,27,40,57,78,94,102,92,72,47,67,88,66,86,99,106,101,93,77,56,39,26,15,24,37,54,75,87,63,42,25,12,3,7,14,23,36,53,74,91,76,51,30,46,65,85,98,105,109,108,103,95,80,60,43,28,41,58,79,69,48,31,18,9,4)(110,111,113,115,117,119,121,123,125,127,129,131,133,135,137,139,140,138,136,134,132,130,128,126,124,122,120,118,116,114,112)Polycyclic presentation:
2^2,31,109|a^2=b,c^a=c^30,d^a=d^76,d^b=d^108871 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(13516, 7);