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Cayley diagram

C4×#272 #6230

order216
isoclinismisoclinic class of #272
automorphismAut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C3⋊D3 · Out(G) = C22×S4
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2

Aliases

direct productC4×#272

Splittings

FittingC4×H(3,𝔽3) ⋊ C2
derived, SylowH(3,𝔽3) ⋊ C2×C4

Subgroups

subgroupG/H
center, hypercenterC3×C4C3⋊D3
commutator, hypocenterH(3,𝔽3)C2×C4
Frattini, socleC2×C3C2×C3⋊D3
FittingC4×H(3,𝔽3)C2
prime radicalC2×#272C2
cosocleC2×H(3,𝔽3)C22
LCSG ▷ H(3,𝔽3)
UCSC1 ◁ C3×C4

88 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

derived subgroup of#13146459, #13146463, #13146464, …
maximal subgroup of#88434, #88435, C8×#272, …
maximal normal subgroup of#88434, #88435, C8×#272, …
maximal quotient ofC8×#272, #88527, C4×#1084, …
automorphism group ofC5×3−1+2, C2×C5×3−1+2
outer automorphism group ofC5×C9⋊C9

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 67);