Finite Group DatabasebetaFinite Group Databasebeta| order | 416 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C13 ⋊ C4×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D13 |
| commutator | C2×C13 | C22×C4 |
| Fitting | C13×C4⋊C4 | C2 |
| socle, prime radical, cosocle | C22×C13 | C23 |
| hypercenter | C4⋊C4 | D13 |
| hypocenter | C13 | C4×Q8 |
| LCS | G ▷ C2×C13 ▷ C13 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
70 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 416 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,4,6,8,10,12,13,11,9,7,5,3)(14,16,17,15)(20,24)(22,25);(18,20,21,24)(19,22,23,25);(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15,17,16)(18,19,21,23)(20,22,24,25)Polycyclic presentation:
2^5,13|a^2=e,b^2=d,c^2=de,f^a=f^12,c^b=cd80 conjugacy classes — view the full character table.
| maximal subgroup of | #11718244, #11718418, #11718419, … |
| maximal normal subgroup of | #11718244, #11718418, #11718419, … |
| maximal quotient of | #11718307, #11718310, #11718313, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(416, 108);