Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of #6104 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C13×#41 ⋊ C2 |
| Sylow | C13 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22⋊D13 |
| commutator, cosocle | C4×C13 | C23 |
| Frattini | C4 | C22×D13 |
| Fitting | C13×#41 | C2 |
| socle | C2×C13 | C2×D4 |
| hypercenter | #41 | D13 |
| hypocenter | C13 | #135 |
| LCS | G ▷ C4×C13 ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #41 | |
62 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 208 |
A minimal faithful permutation representation — 3 generators on 29 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(15,20)(17,25)(19,26)(21,22)(23,29)(27,28);(14,16,18,24)(15,19,22,28)(17,23,25,29)(20,26,21,27);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,19)(17,20)(18,22)(21,25)(23,26)(24,28)(27,29)Polycyclic presentation:
2^5,13|b^2=e,d^2=e,c^a=cd,d^a=de,f^a=f^12,d^c=de74 conjugacy classes — view the full character table.
| maximal subgroup of | #11718484, #11718485, #11718805, … |
| maximal normal subgroup of | #11718484, #11718485, #11718805, … |
| maximal quotient of | #11718657, #11718659, #11718663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 171);