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(C22⋊ΓL(1,𝔽8))⋊D7 #11924342

order1176
isoclinismisoclinic class of (C22⋊ΓL(1,𝔽8))⋊D7
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C2

Aliases

unambiguous semidirect product(C22⋊ΓL(1,𝔽8))⋊D7

Splittings

Fitting, socle, derivedC22×C72 ⋊ C2×C3
HallC22×C7⋊D7 ⋊ C3
SylowC72 ⋊ C2×A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC22×C72C2×C3
LCSG ▷ C22×C72
UCSC1

80 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degreeμt(G) = 42

Polycyclic presentation:

2,3,2^2,7^2|e^a=e^6,f^a=f^6,c^b=cd,d^b=c,e^b=e^4

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4×ΓL(1,𝔽8)⋊D7, #72222158, #72222160, …
maximal normal subgroup ofA4×ΓL(1,𝔽8)⋊D7, #72222158, #72222160, …
maximal quotient ofC2×(C22⋊ΓL(1,𝔽8))⋊D7, #72222272

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1176, 257);