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C2×C3×C4×GL(2,𝔽3) #11922607

order1152
isoclinismisoclinic class of #226
automorphism?
homologyAbelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23

Aliases

direct productC2×C3×C4×GL(2,𝔽3)

Splittings

FittingC2×C3×C4×Q8 ⋊ D3
derivedSL(2,𝔽3) ⋊ C22×C3×C4

Subgroups

subgroupG/H
center, hypercenterC22×C3×C4S4
commutator, hypocenterSL(2,𝔽3)C22×C3×C4
FrattiniC22C22×C3×S4
FittingC2×C3×C4×Q8D3
socleC23×C3C2×S4
prime radicalC22×C3×GL(2,𝔽3)C2
cosocleC2×SL(2,𝔽3)C23×C3
LCSG ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3×C4

801 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 801 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 192

Conjugacy classes & character table

192 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C3×C4×AGL(2,𝔽3), C2×C3×AGL(1,𝔽5)×GL(2,𝔽3)
maximal quotient ofC2×C3×C4×AGL(2,𝔽3), C2×C3×AGL(1,𝔽5)×GL(2,𝔽3)
automorphism group ofC32×C5×C7, C32×C4×C13
outer automorphism group ofC32×C5×C7, C32×C4×C13

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1152, 156618);