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C4×C7×GL(2,𝔽3) #13057112

order1344
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C2×C4×C7 · Schur multiplier (H2) = C2

Aliases

direct productC4×C7×GL(2,𝔽3)

Splittings

FittingC4×C7×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C2×C4×C7

Subgroups

subgroupG/H
center, hypercenterC2×C4×C7S4
commutator, hypocenterSL(2,𝔽3)C2×C4×C7
FrattiniC22C2×C7×S4
FittingC4×C7×Q8D3
socleC22×C7C2×S4
prime radicalC2×C7×GL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C22×C7
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4×C7

176 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degreeμt(G) = 224

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC7×AGL(1,𝔽5)×GL(2,𝔽3), C4×ΓL(1,𝔽8)×GL(2,𝔽3)
maximal normal subgroup ofC4×ΓL(1,𝔽8)×GL(2,𝔽3)
maximal quotient ofC7×AGL(1,𝔽5)×GL(2,𝔽3)
automorphism group ofC32×C29
outer automorphism group ofC32×C29

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1344, 6401);