Finite Group DatabasebetaFinite Group Databasebeta| order | 368 |
| isoclinism | isoclinic class of #67272 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C23 ⋊ Q16 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22⋊D23 |
| commutator, cosocle | C4×C23 | C22 |
| Frattini | C4 | C2×D23 |
| Fitting | Q8×C23 | C2 |
| socle, prime radical | C2×C23 | D4 |
| hypercenter | Q8 | D23 |
| hypocenter | C23 | Q16 |
| LCS | G ▷ C4×C23 ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
18 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 368 |
A minimal faithful permutation representation — 2 generators on 39 points:
(1,2,4,6,8,10,12,14,16,18,20,22,23,21,19,17,15,13,11,9,7,5,3)(24,26,28,34)(25,29,31,38)(27,36,37,33)(30,39,35,32);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25,28,31)(26,32,34,39)(27,35,37,30)(29,36,38,33)Polycyclic presentation:
2^4,23|a^2=d,b^2=d,c^2=d,b^a=bc,c^a=cd,e^a=e^22,c^b=cd62 conjugacy classes — view the full character table.
| maximal subgroup of | #10625086, #10625087, Q16×D23, … |
| maximal normal subgroup of | #10625086, #10625087, Q16×D23, … |
| maximal quotient of | #10624989, #10624992, #10625017, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(368, 17);