Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of Dic54 |
| automorphism | Aut(G) = C22×AGL(1,ℤ/27) (order 1944) · Inn(G) = C2×D27 · Out(G) = C2×C9 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C22×C27 ⋊ C2 |
| Sylow | C27 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D27 |
| commutator, cosocle | C2×C27 | C22 |
| Frattini | C2×C9 | C2×D3 |
| Fitting | C22×C27 | C2 |
| socle | C2×C3 | C2×D9 |
| hypercenter | C22 | D27 |
| hypocenter | C27 | D4 |
| LCS | G ▷ C2×C27 ▷ C27 | |
| UCS | C1 ◁ C2 ◁ C22 | |
32 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,25,24,10,8,5,22,20,23,4,7,26,9,19,14,3,27,6,12,17,13,21,2,18,11,16,15)(29,31);(2,5)(3,9)(4,12)(6,7)(8,18)(10,11)(13,20)(14,19)(15,25)(16,24)(17,23)(21,22)(26,27)(28,29)(30,31)Polycyclic presentation:
2^3,3^3|d^3=e^2f,e^3=f^2,b^a=bc,d^a=d^2e,e^a=e^2f,f^a=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #88399, D4×D27, #88401, … |
| maximal normal subgroup of | #88399, D4×D27, #88401, … |
| maximal quotient of | Dic27⋊C4, (C2×D27)⋊C4, #88368, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 8);