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Cayley diagram

C2×C22⋊C27 #6185

order216
isoclinismisoclinic class of A4
automorphismAut(G) = C9×S4 (order 216) · Inn(G) = A4 · Out(G) = C2×C9
homologyAbelianization (H1) = C2×C27 · Schur multiplier (H2) = C2

Aliases

direct productC2×C22⋊C27
unambiguous semidirect productC23⋊C27

Splittings

derivedC22 ⋊ C2×C27
SylowC23 ⋊ C27

Subgroups

subgroupG/H
center, hypercenterC2×C9A4
commutator, hypocenterC22C2×C27
FrattiniC9C2×A4
FittingC23×C9C3
socle, prime radicalC23×C3C9
cosocleC22×C9C2×C3
LCSG ▷ C22
UCSC1 ◁ C2×C9

28 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 33
transitive degreeμt(G) = 54

A minimal faithful permutation representation — 2 generators on 33 points:

(1,2)(4,5,6)(7,12,24,15,27,26,29,8,17,10,20,30,23,32,11,14,13,25,16,28,33,9,19,18,22,21,31);(3,6)(4,5)

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

derived subgroup of(C22⋊C27)⋊Q8, C4⋊((C22⋊C27)⋊C2), #11720654, …
maximal subgroup of(C22⋊C27)⋊C4, C4×C22⋊C27, C2×(C22⋊C27)⋊C2, …
maximal normal subgroup of(C22⋊C27)⋊C4, C4×C22⋊C27, C2×(C22⋊C27)⋊C2, …
maximal quotient ofC4×C22⋊C27, C2×Q8⋊C27, #41⋊C27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(216, 22);