Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #67138 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C3 |
No aliases for this group.
| Hall | C32×C5 ⋊ D4 |
| Sylow | C32 ⋊ D20 |
| Sylow | C5 ⋊ #638 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D5×C3⋊D3 |
| commutator, Fitting, socle, cosocle | C2×C32×C5 | C22 |
| hypocenter | C32×C5 | D4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
96 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,5,3)(6,7,8,9)(11,12)(14,15);(2,3)(4,5)(7,9)(13,14,15);(10,11,12)Polycyclic presentation:
2^3,3^2,5|b^2=c,b^a=bc,f^a=f^4,d^b=d^2,e^b=e^245 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1202, #10624027, #10624041, … |
| maximal normal subgroup of | D3×#1202, #10624027, #10624041, … |
| maximal quotient of | #10623758, #10623762, #10623763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 69);