Finite Group DatabasebetaFinite Group Databasebeta| order | 112 |
| isoclinism | isoclinic class of Dic14 |
| automorphism | Aut(G) = C2×D4×AGL(1,𝔽7) (order 672) · Inn(G) = C2×D7 · Out(G) = C23×C3 |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ M16 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D7 |
| commutator, socle | C2×C7 | C2×C4 |
| Fitting | C2×C4×C7 | C2 |
| prime radical | C22×C7 | C4 |
| cosocle | C4×C7 | C22 |
| hypercenter | C2×C4 | D7 |
| hypocenter | C7 | M16 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4,6,7,5,3)(9,12)(13,15);(2,3)(4,5)(6,7)(8,9,10,13,11,12,14,15)Polycyclic presentation:
2^4,7|a^2=c,c^2=d,b^a=bd,e^a=e^634 conjugacy classes — view the full character table.
| maximal subgroup of | #6379, #6393, #6396, … |
| maximal normal subgroup of | #6379, #6393, #6396, … |
| maximal quotient of | #6377, C4⋊(C7⋊C8), C22⋊(C7⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 9);