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Cayley diagram

C22×D19⋊C3 #90724

order456
isoclinismisoclinic class of D19⋊C3
automorphism?
homologyAbelianization (H1) = C23×C3 · Schur multiplier (H2) = C23

Aliases

direct productC22×D19⋊C3
unambiguous semidirect productD19⋊(C22×C3)
unambiguous semidirect product(C19⋊C3)⋊C23
unambiguous semidirect product(C2×C19⋊C3)⋊C22

Splittings

Fitting, socleC22×C19 ⋊ C2×C3
derived, SylowC19 ⋊ C23×C3
HallC22×D19 ⋊ C3
HallC19⋊C3 ⋊ C23

Subgroups

subgroupG/H
center, hypercenterC22D19⋊C3
commutator, cosocle, hypocenterC19C23×C3
FrattiniC1G
Fitting, socleC22×C19C2×C3
LCSG ▷ C19
UCSC1 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 23
transitive degreeμt(G) = 76

A minimal faithful permutation representation — 3 generators on 23 points:

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

Conjugacy classes & character table

36 conjugacy classes — view the full character table.

Relationships

maximal subgroup of(C2×D19⋊C3)⋊C4, C2×C4×D19⋊C3, C2×C4⋊(D19⋊C3), …
maximal normal subgroup of(C2×D19⋊C3)⋊C4, C2×C4×D19⋊C3, C2×C4⋊(D19⋊C3), …
maximal quotient ofC2×(C19⋊C3)⋊Q8, C2×C4×D19⋊C3, C2×C4⋊(D19⋊C3), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 44);