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Cayley diagram

C4×C22⋊H(3,𝔽3) #88686

order432
isoclinismisoclinic class of C9⋊A4
automorphism?
homologyAbelianization (H1) = C32×C4 · Schur multiplier (H2) = C2×C32

Aliases

direct productC4×C22⋊H(3,𝔽3)
unambiguous semidirect product(C22×C4)⋊H(3,𝔽3)
unambiguous semidirect product(C32×C4)⋊A4
unambiguous semidirect product(C3×C4×A4)⋊C3

Splittings

FittingC22×C32×C4 ⋊ C3
SylowC22×C4 ⋊ H(3,𝔽3)

Subgroups

subgroupG/H
centerC3×C4C3×A4
commutatorC22×C3C32×C4
FrattiniC2×C3C2×C3×A4
FittingC22×C32×C4C3
socle, cosocleC23×C3C2×C32
prime radicalC2×C22⋊H(3,𝔽3)C2
hypercenterC32×C4A4
hypocenterC22C4×H(3,𝔽3)
LCSG ▷ C22×C3 ▷ C22
UCSC1 ◁ C3×C4 ◁ C32×C4

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 17
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,3,4)(5,8)(6,7)(10,13,14)(12,16,17);(6,7,8)(9,10,12)(11,14,16)(13,17,15)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146287, #13146288, #13146289, …
maximal subgroup of(C3⋊(A4⋊C8))⋊C3, #11720736, (C3⋊(C3⋊C8))⋊A4, …
maximal normal subgroup of(C3⋊(A4⋊C8))⋊C3, #11720736, (C3⋊(C3⋊C8))⋊A4, …
maximal quotient ofC4×Q8⋊H(3,𝔽3), C8×C22⋊H(3,𝔽3), #131⋊H(3,𝔽3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 333);