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Cayley diagram

C22×C22⋊ΓL(1,𝔽8) #66753

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

Aliases

direct productC22×C22⋊ΓL(1,𝔽8)

Splittings

Fitting, socle, HallC24×C7 ⋊ C3
derivedC22×C7 ⋊ C22×C3
SylowC24 ⋊ ΓL(1,𝔽8)
SylowC7 ⋊ C22×A4

Subgroups

subgroupG/H
center, hypercenterC22C22⋊ΓL(1,𝔽8)
commutator, cosocle, hypocenterC22×C7C22×C3
FrattiniC1G
Fitting, socleC24×C7C3
LCSG ▷ C22×C7
UCSC1 ◁ C22

78 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 15 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×Dic7⋊A4, (C22⋊D7)⋊A4, C2×C4×C22⋊ΓL(1,𝔽8), …
maximal normal subgroup ofC2×Dic7⋊A4, (C22⋊D7)⋊A4, C2×C4×C22⋊ΓL(1,𝔽8), …
maximal quotient ofC2×C4×C22⋊ΓL(1,𝔽8), D4×C22⋊ΓL(1,𝔽8), C22×C7⋊SL(2,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 222);