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) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C4×C7 ⋊ C2 |
| Sylow | C7 ⋊ #41 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D7 |
| commutator, socle, cosocle | C2×C7 | C23 |
| Frattini | C2 | C22×D7 |
| Fitting | C2×C4×C7 | C2 |
| hypercenter | C2×C4 | D7 |
| hypocenter | C7 | #41 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×C4 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 56 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,4,6,7,5,3)(9,13)(12,15);(8,10,11,14)(9,12,13,15);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^4,7|b^2=d,c^a=cd,e^a=e^634 conjugacy classes — view the full character table.
| maximal subgroup of | #6379, #6399, #6464, … |
| maximal normal subgroup of | #6379, #6399, #6464, … |
| maximal quotient of | C4×Dic14, #6433, #6435, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(112, 30);