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Cayley diagram

C2×#4296 #66790

order342
isoclinismisoclinic class of #4296
automorphismAut(G) = AGL(1,𝔽19) (order 342) · Inn(G) = #4296 · Out(G) = C2
homologyAbelianization (H1) = C2×C9 · Schur multiplier (H2) = C1

Aliases

direct productC2×#4296

Splittings

Fitting, socle, HallC2×C19 ⋊ C9
derived, SylowC19 ⋊ C2×C9

Subgroups

subgroupG/H
center, hypercenterC2#4296
commutator, hypocenterC19C2×C9
FrattiniC1G
Fitting, socleC2×C19C9
prime radicalC2×C19⋊C3C3
cosocleC19⋊C3C2×C3
LCSG ▷ C19
UCSC1 ◁ C2

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 21
transitive degreeμt(G) = 38

A minimal faithful permutation representation — 2 generators on 21 points:

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

Conjugacy classes & character table

22 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#4296⋊C4, C4×#4296, C2×AGL(1,𝔽19), …
maximal normal subgroup of#4296⋊C4, C4×#4296, C2×AGL(1,𝔽19), …
maximal quotient ofC4×#4296, C22×#4296, C2×#10587020, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(342, 8);