Finite Group Databasebeta

← Prev (#975) Next (#977) →

Cayley diagram

C3×AGL(1,ℤ/8) #976

order96
nilpotency class3
isoclinismisoclinic class of AGL(1,ℤ/8)
automorphismAut(G) = C2×D42 (order 128) · Inn(G) = C2×D4 · Out(G) = C23
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C22

Aliases

direct productC3×AGL(1,ℤ/8)

Subgroups

subgroupG/H
center, socleC2×C3C2×D4
commutator, Frattini, cosocleC4C23×C3
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C2×C3×C4 ◁ G

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2)(3,5)(4,6)(7,8)(9,10,11);(2,5)(3,8)(6,7);(2,6)(5,7)

Conjugacy classes & character table

33 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of#5198, #5199, #5200, …
maximal subgroup of#5198, #5199, #5200, …
maximal normal subgroup of#5198, #5199, #5200, …
maximal quotient ofC3×#417, C3×#418, C3×#428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 183);