Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of #272 |
| automorphism | Aut(G) = C2×AGL(2,𝔽3) (order 864) · Inn(G) = C3⋊D3 · Out(G) = C2×S4 |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C2×C3 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C4 |
| Fitting, prime radical, cosocle | C2×H(3,𝔽3) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C2×C3 | |
33 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 36 |
A minimal faithful permutation representation — 3 generators on 13 points:
(2,4)(5,9)(6,8)(10,11,12,13);(2,5,6)(4,8,9);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^2,3^3|a^2=b,c^a=c^2,d^a=d^2,d^c=de20 conjugacy classes — view the full character table.
| derived subgroup of | #88587, #88588, #88589, … |
| maximal subgroup of | #6188, #6196, ((C3⋊D3)⋊C3)⋊C4, … |
| maximal normal subgroup of | #6188, #6196, ((C3⋊D3)⋊C3)⋊C4, … |
| maximal quotient of | #6180, C2×#1084, #66328, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 11);