Finite Group DatabasebetaFinite Group Databasebeta| order | 288 |
| isoclinism | isoclinic class of #63545 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C32 ⋊ Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #3713 |
| commutator, Fitting, cosocle | C32×C8 | C22 |
| Frattini, hypercenter | C8 | D32 |
| socle, prime radical | C2×C32 | D8 |
| hypocenter | C32 | Q32 |
| LCS | G ▷ C32×C8 ▷ C32×C4 ▷ C2×C32 ▷ C32 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
53 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 38 points:
(1,3,6,14)(2,7,10,24)(4,16,19,32)(5,21,17,13)(8,26,27,31)(9,28,20,23)(11,29,30,15)(12,25,22,18)(34,35)(36,37,38);(1,2,6,10)(3,11,14,30)(4,15,19,29)(5,20,17,9)(7,22,24,12)(8,25,27,18)(13,31,21,26)(16,28,32,23)(33,34,35)(37,38)Polycyclic presentation:
2^5,3^2|a^2=e,b^2=e,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,g^a=g^2,c^b=cd,d^b=de,f^b=f^233 conjugacy classes — view the full character table.
| maximal subgroup of | #10590025, #10590026, #10590240, … |
| maximal normal subgroup of | #10590025, #10590026, #10590240, … |
| maximal quotient of | #10589117, #10589140, C2×#63550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(288, 198);