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Cayley diagram

C22×Dic27 #88404

order432
isoclinismisoclinic class of D27
automorphism?
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C23

Aliases

direct productC22×Dic27

Splittings

derived, SylowC27 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23D27
commutator, hypocenterC27C22×C4
FrattiniC2×C9C22×D3
FittingC23×C27C2
socle, prime radicalC23×C3D9
cosocleC2×C27C23
LCSG ▷ C27
UCSC1 ◁ C23

108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 35
transitive degreeμt(G) = 432

A minimal faithful permutation representation — 3 generators on 35 points:

(1,2)(3,4)(5,6,7,8)(10,13)(11,17)(12,20)(14,15)(16,26)(18,19)(21,28)(22,27)(23,33)(24,32)(25,31)(29,30)(34,35);(3,4)(9,28,22,29,33,31,11,10,32,12,35,26,18,15,14,19,16,34,20,24,13,17,25,23,30,27,21);(1,2)

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC27⋊#95, #11720126, #11720127, …
maximal normal subgroup ofC27⋊#95, #11720126, #11720127, …
maximal quotient ofC22×C27⋊C8, C2×#88363, C2×C4×Dic27, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 51);