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Cayley diagram

C7×AGL(1,ℤ/8) #6539

order224
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) = C2×C3×D42 (order 384) · Inn(G) = C2×D4 · Out(G) = C23×C3
homologyAbelianization (H1) = C23×C7 · Schur multiplier (H2) = C22

Aliases

direct productC7×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC2×C7C2×D4
commutator, Frattini, cosocleC4C23×C7
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C7 ◁ C2×C4×C7 ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 56

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

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#89970, #89971, #89972, …
maximal subgroup of#89970, #89971, #89972, …
maximal normal subgroup of#89970, #89971, #89972, …
maximal quotient ofC7×#417, C7×#418, C7×#428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(224, 171);