Finite Group DatabasebetaFinite Group Databasebeta| order | 34560 |
| isoclinism | isoclinic class of #72227492 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C3 | #72209614 |
| commutator, hypocenter | G | C1 |
| Fitting, socle, cosocle | C25×C3 | A6 |
| LCS | G | |
| UCS | C1 ◁ C2×C3 | |
393 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 393 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 114 |
| transitive degree | μt(G) = 720 |
A minimal faithful permutation representation — 2 generators on 114 points:
(1,10,2,14,3,15,8,13)(4,12,9,6,7,11,16,5)(17,70,34,57,21,93,20,78)(18,69,91,48,25,75,60,82)(19,39,59,94,29,61,44,81)(22,52,66,37,36,30,87,67)(23,53,49,90,42,73,71,79)(24,41,62,80,45,63,32,46)(26,86,50,74,47,72,65,40)(27,92,85,31,33,89,95,58)(28,38,96,83,51,68,88,54)(35,55,84,43,64,76,56,77)(97,106,105,104,98,103,111,110,99,109,108,107)(100,112,101,114,102,113);(2,4)(3,5)(7,10)(8,14)(11,16)(13,15)(17,18,21,25)(19,27,29,33)(20,31,34,58)(22,24,36,45)(23,40,42,74)(26,48,47,82)(28,50,51,65)(30,53,52,73)(32,60,62,91)(35,44,64,59)(37,69,67,75)(38,70,68,93)(39,72,61,86)(41,76,63,55)(43,49,77,71)(46,81,80,94)(54,85,83,95)(56,57,84,78)(66,92,87,89)(79,88,90,96)(97,98,99)(100,102,101)(103,112,106,114,109,113)(104,111,107,105,110,108)55 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,10,2,14,3,15,8,13)(4,12,9,6,7,11,16,5)(17,70,34,57,21,93,20,78)(18,69,91,48,25,75,60,82)(19,39,59,94,29,61,44,81)(22,52,66,37,36,30,87,67)(23,53,49,90,42,73,71,79)(24,41,62,80,45,63,32,46)(26,86,50,74,47,72,65,40)(27,92,85,31,33,89,95,58)(28,38,96,83,51,68,88,54)(35,55,84,43,64,76,56,77)(97,106,105,104,98,103,111,110,99,109,108,107)(100,112,101,114,102,113), (2,4)(3,5)(7,10)(8,14)(11,16)(13,15)(17,18,21,25)(19,27,29,33)(20,31,34,58)(22,24,36,45)(23,40,42,74)(26,48,47,82)(28,50,51,65)(30,53,52,73)(32,60,62,91)(35,44,64,59)(37,69,67,75)(38,70,68,93)(39,72,61,86)(41,76,63,55)(43,49,77,71)(46,81,80,94)(54,85,83,95)(56,57,84,78)(66,92,87,89)(79,88,90,96)(97,98,99)(100,102,101)(103,112,106,114,109,113)(104,111,107,105,110,108));