Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| isoclinism | isoclinic class of #6095 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C8×C13 ⋊ C2 |
| Sylow | C13 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D52 |
| commutator, cosocle | C4×C13 | C23 |
| Frattini | C4 | C22×D13 |
| Fitting | C2×C8×C13 | C2 |
| socle | C2×C13 | C2×D4 |
| hypercenter | C2×C8 | D13 |
| hypocenter | C13 | #135 |
| LCS | G ▷ C4×C13 ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
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)(14,17,18,25,19,26,16,23)(15,21,20,24,22,28,27,29);(14,16,19,18)(15,20,22,27)(17,23,26,25)(21,24,28,29);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,20)(17,24)(18,27)(19,22)(21,25)(23,28)(26,29)Polycyclic presentation:
2^5,13|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^12110 conjugacy classes — view the full character table.
| maximal subgroup of | #11718080, #11718089, #11718102, … |
| maximal normal subgroup of | #11718080, #11718089, #11718102, … |
| maximal quotient of | #11718344, C4×#6095, C4×D104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 125);