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Cayley diagram

C22⋊(C22⋊ΓL(1,𝔽8)) #66755

order336
isoclinismisoclinic class of C22⋊(C22⋊ΓL(1,𝔽8))
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C24

Aliases

unambiguous semidirect productC22⋊(C22⋊ΓL(1,𝔽8))
unambiguous semidirect productC7⋊(C22⋊A4)

Splittings

Fitting, socle, derived, HallC24×C7 ⋊ C3
SylowC24 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C22⋊A4

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC24×C7C3
LCSG ▷ C24×C7
UCSC1

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

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

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

Polycyclic presentation:

3,2^4,7|b^a=c,c^a=bc,d^a=e,e^a=de,f^a=f^2

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10620804, #120⋊ΓL(1,𝔽8), D7⋊(C22⋊A4), …
maximal normal subgroup of#10620804, #120⋊ΓL(1,𝔽8), D7⋊(C22⋊A4), …
maximal quotient ofC22⋊(C7⋊SL(2,𝔽3)), C7⋊#997, C2×C22⋊(C22⋊ΓL(1,𝔽8)), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 224);