Finite Group DatabasebetaFinite Group Databasebeta| order | 360 |
| isoclinism | isoclinic class of #4379 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C3 |
No aliases for this group.
| derived, Hall | C32×C5 ⋊ C8 |
| Sylow | C32 ⋊ #168 |
| Sylow | C5 ⋊ #622 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #4379 |
| commutator, hypocenter | C32×C5 | C8 |
| Fitting, socle, prime radical | C2×C32×C5 | C4 |
| cosocle | C5×C3⋊Dic3 | C2 |
| LCS | G ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3)(4,5)(6,7)(8,9,11,10)(12,13,14,16,15,17,18,19);(1,2,4,5,3)(6,8,11)Polycyclic presentation:
2^3,3^2,5|a^2=b,b^2=c,d^a=d^2e,e^a=de,f^a=f^4,d^b=d^2,e^b=e^236 conjugacy classes — view the full character table.
| maximal subgroup of | #10623955, D5×#622, #10623988, … |
| maximal normal subgroup of | #10623955, D5×#622, #10623988, … |
| maximal quotient of | #10623724, C2×#67126, #11761728, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(360, 56);