Finite Group DatabasebetaFinite Group Databasebeta| order | 108 |
| isoclinism | isoclinic class of C9⋊A4 |
| automorphism | Aut(G) = A4⋊#272 (order 648) · Inn(G) = C3×A4 · Out(G) = C3⋊D3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C22 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×A4 |
| commutator, socle, cosocle | C22×C3 | C32 |
| Fitting, prime radical | C22×C32 | C3 |
| hypercenter | C32 | A4 |
| hypocenter | C22 | 3−1+2 |
| LCS | G ▷ C22×C3 ▷ C22 | |
| UCS | C1 ◁ C3 ◁ C32 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 18 |
A minimal faithful permutation representation — 2 generators on 13 points:
(1,4)(2,3)(6,9,10)(8,12,13);(2,3,4)(5,6,8,7,10,12,11,9,13)Polycyclic presentation:
3^3,2^2|a^3=c,b^a=bc,d^a=e,e^a=de20 conjugacy classes — view the full character table.
| maximal subgroup of | ((C22⋊C9)⋊C2)⋊C3, C2×#1094, #66357, … |
| maximal normal subgroup of | ((C22⋊C9)⋊C2)⋊C3, C2×#1094, #66357, … |
| maximal quotient of | #6204, C2×#1094, (C22⋊C9)⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(108, 21);