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Cayley diagram

Q8×D23 #67291

order368
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productQ8×D23

Splittings

FittingQ8×C23 ⋊ C2
SylowC23 ⋊ C2×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×D23
commutator, socle, cosocleC2×C23C23
FittingQ8×C23C2
prime radicalC2×D23C22
hypercenterQ8D23
hypocenterC23C2×Q8
LCSG ▷ C2×C23 ▷ C23
UCSC1 ◁ C2 ◁ Q8

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 184

A minimal faithful permutation representation — 3 generators on 31 points:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×D23, #10625086, Q16×D23, …
maximal normal subgroup ofSD16×D23, #10625086, Q16×D23, …
maximal quotient of#10625058, #10625059, #10625060, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(368, 33);