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Cayley diagram

C2×#1088 #6263

order216
isoclinismisoclinic class of #1088
automorphismAut(G) = C2×AΓL(1,𝔽9) (order 288) · Inn(G) = #157 · Out(G) = C23
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC2×#1088

Splittings

FittingC2×H(3,𝔽3) ⋊ C4
derived, SylowH(3,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, socle, hypercenterC2×C3#157
commutator, hypocenterH(3,𝔽3)C2×C4
FrattiniC3C2×#157
FittingC2×H(3,𝔽3)C4
prime radicalC2×#272C2
cosocle#272C22
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C2×C3

44 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 20
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 2 generators on 20 points:

(3,4)(5,9,10,17)(6,8)(7,12,14,20)(11,18,19,15)(13,16);(1,2)(3,10,11)(4,8,16)(5,13,12)(6,19,20)(7,18,9)

Conjugacy classes & character table

28 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146168, #13146170, #13146176, …
maximal subgroup of#88590, (((C3⋊D3)⋊C3)⋊C2)⋊C4, #88592, …
maximal normal subgroup of#88590, (((C3⋊D3)⋊C3)⋊C2)⋊C4, #88592, …
maximal quotient of#88626, #88627, C4×#1088, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 100);