Finite Group DatabasebetaFinite Group Databasebeta| order | 12956 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C79⋊#4223 |
| Fitting, socle, derived, Hall | C41×C79 ⋊ C4 |
| Sylow | C79 ⋊ #4223 |
| Sylow | C41 ⋊ Dic79 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C41×C79 | C4 |
| prime radical, cosocle | C79×D41 | C2 |
| LCS | G ▷ C41×C79 | |
| UCS | C1 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 120 |
| transitive degree | μt(G) = 3239 |
A minimal faithful permutation representation — 2 generators on 120 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31)(32,33)(34,35)(36,37)(38,39)(40,41)(42,43)(44,45)(46,47)(48,49)(50,51)(52,53)(54,55)(56,57)(58,59)(60,61)(62,63)(64,65)(66,67)(68,69)(70,71)(72,73)(74,75)(76,77)(78,79)(81,82,83,85)(84,87,88,94)(86,91,92,90)(89,96,97,107)(93,100,101,103)(95,104,105,99)(98,109,110,114)(102,112,113,119)(106,116,117,115)(108,120,118,111);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,74,76,78,79,77,75,73,71,69,67,65,63,61,59,57,55,53,51,49,47,45,43,41,39,37,35,33,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(80,81,84,89,98,109,113,101,92,85,90,99,111,112,96,108,117,105,94,103,115,116,100,87,95,106,118,107,119,120,104,91,82,86,93,102,114,110,97,88,83)Polycyclic presentation:
2^2,41,79|a^2=b,c^a=c^32,d^a=d^78,c^b=c^40872 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12956, 7);