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Cayley diagram

D7×C4⋊C4 #6454

order224
automorphismAut(G) = C22×#120×AGL(1,𝔽7) (order 5376) · Inn(G) = C22×D7 · Out(G) = C22×C3×D4
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productD7×C4⋊C4

Splittings

FittingC7×C4⋊C4 ⋊ C2
SylowC7 ⋊ C2×C4⋊C4

Subgroups

subgroupG/H
center, FrattiniC22C22×D7
commutatorC2×C7C22×C4
FittingC7×C4⋊C4C2
socle, cosocleC22×C7C23
prime radicalC22×D7C22
hypercenterC4⋊C4D7
hypocenterC7C2×C4⋊C4
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4⋊C4

92 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 112

A minimal faithful permutation representation — 3 generators on 15 points:

(5,6,7,8)(9,10,12,14,15,13,11);(1,2,3,4)(6,8);(10,11)(12,13)(14,15)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×#102, #89543, #89546, …
maximal normal subgroup ofD7×#102, #89543, #89546, …
maximal quotient of#89419, #89421, #89422, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 86);