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Cayley diagram

C2×((C3⋊D3)⋊C3)⋊C2 #6265

order216
isoclinismisoclinic class of ((C3⋊D3)⋊C3)⋊C2
automorphismAut(G) = #11722315 (order 864) · Inn(G) = ((C3⋊D3)⋊C3)⋊C2 · Out(G) = D4
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

BorelB(3,𝔽3)
direct productC2×((C3⋊D3)⋊C3)⋊C2
unambiguous semidirect product((C3⋊D3)⋊C3)⋊C22
unambiguous semidirect product#272⋊C22

Splittings

FittingC2×H(3,𝔽3) ⋊ C22
derived, SylowH(3,𝔽3) ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC2((C3⋊D3)⋊C3)⋊C2
commutator, cosocle, hypocenterH(3,𝔽3)C23
FrattiniC3C2×D32
FittingC2×H(3,𝔽3)C22
socleC2×C3D32
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2

122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 11
transitive degreeμt(G) = 18

A minimal faithful permutation representation — 3 generators on 11 points:

(1,2)(4,10,8,6,7,11)(5,9);(3,4,6)(5,7,10,9,8,11);(1,2)

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(((C3⋊D3)⋊C3)⋊C2)⋊C4, C4×((C3⋊D3)⋊C3)⋊C2, #88654, …
maximal normal subgroup of(((C3⋊D3)⋊C3)⋊C2)⋊C4, C4×((C3⋊D3)⋊C3)⋊C2, #88654, …
maximal quotient of((C3⋊D3)⋊C3)⋊Q8, #88648, #88649, …
automorphism group of(C3⋊Dic3)⋊C3, C2×(C3⋊D3)⋊C3

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 102);