Finite Group DatabasebetaFinite Group Databasebeta| order | 356 |
| isoclinism | isoclinic class of #67063 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, socle, derived, Sylow | C89 ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, hypocenter | C89 | C4 |
| prime radical, cosocle | D89 | C2 |
| LCS | G ▷ C89 | |
| UCS | C1 | |
6 conjugacy classes of subgroups.
| minimal degree | μ(G) = 89 |
| transitive degree | μt(G) = 89 |
A minimal faithful permutation representation — 2 generators on 89 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)(53,72,73,83)(57,76,77,81)(61,80,78,79)(65,82,74,75)(67,84,70,71)(86,88,89,87);(1,2,5,10,19,32,49,70,66,83,80,55,34,17,29,44,61,77,56,39,26,15,24,37,54,75,68,86,88,84,63,42,25,12,3,7,14,23,36,53,74,64,45,62,81,76,51,30,46,65,73,52,35,22,13,6,11,20,33,50,71,87,89,85,82,59,38,21,8,16,27,40,57,78,60,43,28,41,58,79,72,47,67,69,48,31,18,9,4)Polycyclic presentation:
2^2,89|a^2=b,c^a=c^34,c^b=c^8826 conjugacy classes — view the full character table.
| maximal subgroup of | #10623505, C2×#67063, C3×#67063, … |
| maximal normal subgroup of | #10623505, C2×#67063, C3×#67063, … |
| maximal quotient of | #10623496, C2×#67063, C3×#67063, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(356, 3);