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Cayley diagram

Q16×D13 #88171

order416
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productQ16×D13

Splittings

FittingC13×Q16 ⋊ C2
SylowC13 ⋊ C2×Q16

Subgroups

subgroupG/H
centerC2D4×D13
commutator, cosocleC4×C13C23
FrattiniC4C22×D13
FittingC13×Q16C2
socleC2×C13C2×D4
prime radicalC2×D13D4
hypercenterQ16D13
hypocenterC13C2×Q16
LCSG ▷ C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2 ◁ C4 ◁ Q16

60 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 29
transitive degreeμt(G) = 208

A minimal faithful permutation representation — 3 generators on 29 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11718280, #11718282, D13×SD32, …
maximal normal subgroup of#11718280, #11718282, D13×SD32, …
maximal quotient of#11718451, #11718453, #11718454, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(416, 138);