Finite Group DatabasebetaFinite Group Databasebeta| order | 400 |
| isoclinism | isoclinic class of Dic50 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C25 ⋊ C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×D25 |
| commutator | C2×C25 | C2×C4 |
| Frattini, socle, prime radical | C22×C5 | C2×D5 |
| Fitting | C2×C4×C25 | C2 |
| cosocle | C22×C25 | C22 |
| hypercenter | C2×C4 | D25 |
| hypocenter | C25 | C4⋊C4 |
| LCS | G ▷ C2×C25 ▷ C25 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 33 |
| transitive degree | μt(G) = 400 |
A minimal faithful permutation representation — 2 generators on 33 points:
(1,21,12,24,9,7,14,5,23,4,16,6,22,18,13,8,2,25,10,20,3,15,19,11,17)(26,27,28,29)(31,33);(2,4)(3,7)(5,10)(6,13)(8,16)(9,15)(11,12)(14,20)(17,21)(18,22)(19,24)(23,25)(30,31,32,33)Polycyclic presentation:
2^4,5^2|a^2=d,b^2=c,e^5=f^4,b^a=bd,e^a=e^4f,f^a=f^4106 conjugacy classes — view the full character table.
| maximal subgroup of | C4×Dic50, #11716432, #11716434, … |
| maximal normal subgroup of | C4×Dic50, #11716432, #11716434, … |
| maximal quotient of | #11716355, #11716356, #11716363, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(400, 12);