Finite Group DatabasebetaFinite Group Databasebeta| order | 368 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | D4×C23 ⋊ C2 |
| Sylow | C23 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D23 |
| commutator, socle, cosocle | C2×C23 | C23 |
| Fitting | D4×C23 | C2 |
| hypercenter | D4 | D23 |
| hypocenter | C23 | #41 |
| LCS | G ▷ C2×C23 ▷ C23 | |
| UCS | C1 ◁ C2 ◁ D4 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 184 |
A minimal faithful permutation representation — 3 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)(25,29)(26,30);(24,26,27,30)(25,28,29,31);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,28)(27,29)(30,31)Polycyclic presentation:
2^4,23|b^2=d,c^a=cd,e^a=e^22,c^b=cd65 conjugacy classes — view the full character table.
| maximal subgroup of | #10625082, #10625083, #10625086, … |
| maximal normal subgroup of | #10625082, #10625083, #10625086, … |
| maximal quotient of | #10625048, #10625049, #10625050, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(368, 32);