Finite Group DatabasebetaFinite Group Databasebeta| order | 896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C7×#431 ⋊ C2 |
| Sylow | C7 ⋊ #3207 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D28 |
| commutator, cosocle | C2×C4×C7 | C24 |
| Frattini | C2×C4 | C23×D7 |
| Fitting | C7×#431 | C2 |
| socle | C22×C7 | C22×D4 |
| hypercenter | #431 | D7 |
| hypocenter | C7 | #3207 |
| LCS | G ▷ C2×C4×C7 ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #431 | |
438 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 438 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 448 |
Polycyclic presentation:
2^7,7|b^2=e,d^2=g,f^2=g,b^a=be,c^a=cf,d^a=de,f^a=fg,h^a=h^6,c^b=ce,f^c=fg,h^c=h^6152 conjugacy classes — view the full character table.
| maximal subgroup of | #13250396, #13250398, #13250410, … |
| maximal normal subgroup of | #13250396, #13250398, #13250410, … |
| maximal quotient of | #13363115, #13363501, #13364334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(896, 11515);