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Cayley diagram

C2×#6249 #88882

order432
isoclinismisoclinic class of #6249
automorphismAut(G) = C2×#6249⋊C2 (order 864) · Inn(G) = #6249 · Out(G) = C22
homologyAbelianization (H1) = C2×C8 · Schur multiplier (H2) = C2

Aliases

direct productC2×#6249

Splittings

FittingC2×H(3,𝔽3) ⋊ C8
derived, SylowH(3,𝔽3) ⋊ C2×C8

Subgroups

subgroupG/H
center, hypercenterC2#6249
commutator, hypocenterH(3,𝔽3)C2×C8
FrattiniC3C2×AGL(1,𝔽9)
FittingC2×H(3,𝔽3)C8
socleC2×C3AGL(1,𝔽9)
prime radicalC2×#272C4
cosocle#1088C22
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2

43 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 29 points:

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

Conjugacy classes & character table

26 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#6249⋊C4, (#1088⋊C2)⋊C4, SU(3,𝔽2)⋊C4, …
maximal normal subgroup of#6249⋊C4, (#1088⋊C2)⋊C4, SU(3,𝔽2)⋊C4, …
maximal quotient of#11722287, #11722288, C4×#6249, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 529);