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Cayley diagram

Q8⋊(C19⋊C3) #90703

order456
isoclinismisoclinic class of Q8⋊(C19⋊C3)
automorphism?
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C1

Aliases

unambiguous semidirect productQ8⋊(C19⋊C3)
unambiguous semidirect productC19⋊SL(2,𝔽3)

Splittings

Fitting, derived, HallQ8×C19 ⋊ C3
SylowC19 ⋊ SL(2,𝔽3)
SylowQ8 ⋊ C19⋊C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC2C22⋊(C19⋊C3)
commutator, Fitting, cosocle, hypocenterQ8×C19C3
socleC2×C19A4
LCSG ▷ Q8×C19
UCSC1 ◁ C2

14 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:

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

Polycyclic presentation:

3,2^2,19,2|b^2=e,c^2=e,b^a=c,c^a=bc,d^a=d^7,c^b=ce

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744922, D19⋊SL(2,𝔽3), C2×Q8⋊(C19⋊C3), …
maximal normal subgroup of#11744922, D19⋊SL(2,𝔽3), C2×Q8⋊(C19⋊C3), …
maximal quotient ofC2×Q8⋊(C19⋊C3), Q8⋊#4294, C3×Q8⋊(C19⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 23);