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C7×(C3⋊D3)⋊ΓL(1,𝔽8) #72224663

order2646
isoclinismisoclinic class of (C3⋊D3)⋊ΓL(1,𝔽8)
automorphism?
homologyAbelianization (H1) = C2×C3×C7 · Schur multiplier (H2) = C3

Aliases

direct productC7×(C3⋊D3)⋊ΓL(1,𝔽8)

Splittings

FittingC32×C72 ⋊ C2×C3
derivedC32×C7 ⋊ C2×C3×C7
HallC7×C7⋊H(3,𝔽3) ⋊ C2
SylowC72 ⋊ (C3⋊D3)⋊C3

Subgroups

subgroupG/H
center, hypercenterC7(C3⋊D3)⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC32×C7C2×C3×C7
FrattiniC3D3×C7×ΓL(1,𝔽8)
FittingC32×C72C2×C3
socleC3×C72C3×D3
LCSG ▷ C32×C7
UCSC1 ◁ C7

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 63

A minimal faithful permutation representation — 2 generators on 23 points:

(1,2,3,4,5,6,7)(9,10,11)(12,14,13)(16,21,23,18,19,22)(17,20);(8,9,10,12,11,13,14)(15,16,18)(17,19,22)(20,23,21)

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#72224666, #72224673, C7×D7⋊((C3⋊D3)⋊C3), …
maximal normal subgroup of#72224666, #72224673, C7×D7⋊((C3⋊D3)⋊C3), …
maximal quotient ofC2×C7×(C3⋊D3)⋊ΓL(1,𝔽8)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

gap> G := Group((1,2,3,4,5,6,7)(9,10,11)(12,14,13)(16,21,23,18,19,22)(17,20), (8,9,10,12,11,13,14)(15,16,18)(17,19,22)(20,23,21));