Finite Group DatabasebetaFinite Group Databasebeta| order | 144 |
| automorphism | Aut(G) = C2×#64241 (order 576) · Inn(G) = C2×D32 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C32 ⋊ C22 |
| Sylow | C32 ⋊ C2×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32 |
| commutator, socle, cosocle | C2×C32 | C23 |
| Fitting | C22×C32 | C22 |
| hypercenter | C22 | D32 |
| hypocenter | C32 | C2×D4 |
| LCS | G ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2)(3,4)(5,6,7)(8,9,10);(2,4)(6,7);(2,4)(9,10)Polycyclic presentation:
2^4,3^2|c^a=cd,f^a=f^2,c^b=cd,e^b=e^224 conjugacy classes — view the full character table.
| maximal subgroup of | #63738, #64233, #64241, … |
| maximal normal subgroup of | #63738, #64233, #64241, … |
| maximal quotient of | #63840, #63853, #63865, … |
| holomorph of | Dic3, C2×D3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(144, 154);