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Cayley diagram

C2×Dic34 #63249

order272
isoclinismisoclinic class of Dic34
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productC2×Dic34
unambiguous semidirect product(C2×C17)⋊Q8

Splittings

SylowC17 ⋊ C2×Q8

Subgroups

subgroupG/H
centerC22C2×D17
commutator, cosocleC2×C17C23
FrattiniC2C22×D17
FittingC2×C4×C17C2
socle, prime radicalC22×C17C22
hypercenterC2×C4D17
hypocenterC17C2×Q8
LCSG ▷ C2×C17 ▷ C17
UCSC1 ◁ C22 ◁ C2×C4

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 272

A minimal faithful permutation representation — 3 generators on 27 points:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#10587481, #10587487, #10587495, …
maximal normal subgroup of#10587481, #10587487, #10587495, …
maximal quotient ofC4×Dic34, #10587555, #10587556, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(272, 36);