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Cayley diagram

C2×(C22⋊C27)⋊C2 #88577

order432
isoclinismisoclinic class of (C22⋊C27)⋊C2
automorphism?
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

direct productC2×(C22⋊C27)⋊C2
unambiguous semidirect product(C22⋊C27)⋊C22
unambiguous semidirect product(C2×C22⋊C27)⋊C2

Splittings

derivedC22⋊C27 ⋊ C22

Subgroups

subgroupG/H
center, hypercenterC2(C22⋊C27)⋊C2
commutator, cosocle, hypocenterC22⋊C27C22
FrattiniC9C2×S4
FittingC23×C9D3
socleC23×C3D9
LCSG ▷ C22⋊C27
UCSC1 ◁ C2

79 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:

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

Conjugacy classes & character table

42 conjugacy classes — view the full character table.

Relationships

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 224);