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Cayley diagram

Q8×C19⋊C3 #90701

order456
isoclinismisoclinic class of D4×C19⋊C3
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C1

Aliases

direct productQ8×C19⋊C3

Splittings

Fitting, HallQ8×C19 ⋊ C3
SylowC19 ⋊ C3×Q8

Subgroups

subgroupG/H
center, FrattiniC2C22×C19⋊C3
commutator, socle, cosocleC2×C19C22×C3
FittingQ8×C19C3
prime radicalC2×C19⋊C3C22
hypercenterQ8C19⋊C3
hypocenterC19C3×Q8
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2 ◁ Q8

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 27 points:

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

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#64527⋊C3, #64528⋊C3, SD16×C19⋊C3, …
maximal normal subgroup of#64527⋊C3, #64528⋊C3, SD16×C19⋊C3, …
maximal quotient ofC4⋊C4×C19⋊C3, C2×Q8×C19⋊C3, Q8×#4294, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 21);