Finite Group DatabasebetaFinite Group Databasebeta| order | 232 |
| isoclinism | isoclinic class of #1181 |
| automorphism | Aut(G) = C2×AGL(1,𝔽29) (order 1624) · Inn(G) = #1181 · Out(G) = C2×C7 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | C29 ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #1181 |
| commutator, hypocenter | C29 | C8 |
| Fitting, socle, prime radical | C2×C29 | C4 |
| cosocle | Dic29 | C2 |
| LCS | G ▷ C29 | |
| UCS | C1 ◁ C2 | |
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:
(2,3,4,6)(5,8,9,15)(7,12,13,11)(10,17,18,25)(14,21,22,23)(16,24,19,20)(26,28,29,27)(30,31,32,34,33,35,36,37);(1,2,5,10,19,15,23,17,26,28,24,12,3,7,14,22,13,6,11,20,27,29,25,21,8,16,18,9,4)Polycyclic presentation:
2^3,29|a^2=b,b^2=c,d^a=d^12,d^b=d^2822 conjugacy classes — view the full character table.
| maximal subgroup of | D29⋊C8, #90796, C2×#6599, … |
| maximal normal subgroup of | D29⋊C8, #90796, C2×#6599, … |
| maximal quotient of | #90770, C2×#6599, C3×#6599, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(232, 3);