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Cayley diagram

C2×SU(3,𝔽2) #88884

order432
isoclinismisoclinic class of SU(3,𝔽2)
automorphismAut(G) = #13181783 (order 1728) · Inn(G) = PSU(3,𝔽2) · Out(G) = S4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×SU(3,𝔽2)

Splittings

FittingC2×H(3,𝔽3) ⋊ Q8
SylowH(3,𝔽3) ⋊ C2×Q8

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3PSU(3,𝔽2)
commutator, cosocle#272C23
FrattiniC3C2×PSU(3,𝔽2)
FittingC2×H(3,𝔽3)Q8
prime radicalC2×#272C22
hypocenterH(3,𝔽3)C2×Q8
LCSG ▷ #272 ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3

67 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 3 generators on 29 points:

(1,2)(3,19,9,10,17,7,21,26,6,28,8,14)(4,23,11)(5,25,12,18,27,13,29,22,16,15,24,20);(4,7,8,18)(5,12,13,10)(9,22,23,28)(11,19,21,20)(14,16,24,25)(15,26,27,29);(1,2)

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSU(3,𝔽2)⋊C4, C4×SU(3,𝔽2), C4⋊SU(3,𝔽2), …
maximal normal subgroup ofSU(3,𝔽2)⋊C4, C4×SU(3,𝔽2), C4⋊SU(3,𝔽2), …
maximal quotient of#11722317, C4×SU(3,𝔽2), C4⋊SU(3,𝔽2), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 531);