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Cayley diagram

Q8×D19 #64544

order304
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

direct productQ8×D19

Splittings

FittingQ8×C19 ⋊ C2
SylowC19 ⋊ C2×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×D19
commutator, socle, cosocleC2×C19C23
FittingQ8×C19C2
prime radicalC2×D19C22
hypercenterQ8D19
hypocenterC19C2×Q8
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2 ◁ Q8

38 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 27
transitive degreeμt(G) = 152

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

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofSD16×D19, #10597361, Q16×D19, …
maximal normal subgroup ofSD16×D19, #10597361, Q16×D19, …
maximal quotient of#10597333, #10597334, #10597335, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(304, 33);