Finite Group DatabasebetaFinite Group Databasebeta| order | 368 |
| isoclinism | isoclinic class of #67263 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C8×C23 ⋊ C2 |
| Sylow | C23 ⋊ SD16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D92 |
| commutator, cosocle | C4×C23 | C22 |
| Frattini | C4 | C2×D23 |
| Fitting | C8×C23 | C2 |
| socle | C2×C23 | D4 |
| prime radical | D92 | C2 |
| hypercenter | C8 | D23 |
| hypocenter | C23 | SD16 |
| LCS | G ▷ C4×C23 ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 184 |
A minimal faithful permutation representation — 2 generators on 31 points:
(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,25,26,28,27,29,30,31);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(25,28)(26,30)(29,31)Polycyclic presentation:
2^4,23|b^2=c,c^2=d,b^a=bc,c^a=cd,e^a=e^2295 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#67263, #10625075, #10625079, … |
| maximal normal subgroup of | C2×#67263, #10625075, #10625079, … |
| maximal quotient of | #10624998, #10624999, #10625003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(368, 5);