Finite Group DatabasebetaFinite Group Databasebeta| order | 232 |
| isoclinism | isoclinic class of D29 |
| automorphism | Aut(G) = C22×AGL(1,𝔽29) (order 3248) · Inn(G) = D29 · Out(G) = C23×C7 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C29 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D29 |
| commutator, hypocenter | C29 | C8 |
| Fitting, cosocle | C4×C29 | C2 |
| socle, prime radical | C2×C29 | C4 |
| LCS | G ▷ C29 | |
| UCS | C1 ◁ C4 | |
8 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 232 |
A minimal faithful permutation representation — 2 generators on 37 points:
(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31,32,34,33,35,36,37);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,29,27,25,23,21,19,17,15,13,11,9,7,5,3)Polycyclic presentation:
2^3,29|a^2=b,b^2=c,d^a=d^2864 conjugacy classes — view the full character table.
| maximal subgroup of | #90770, C8×D29, #90772, … |
| maximal normal subgroup of | #90770, C8×D29, #90772, … |
| maximal quotient of | #90768, C2×#6597, C3×#6597, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(232, 1);