Finite Group Databasebeta

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Cayley diagram

C3×D76 #90706

order456
isoclinismisoclinic class of Dic38
automorphism?
homologyAbelianization (H1) = C22×C3 · Schur multiplier (H2) = C2

Aliases

direct productC3×D76
unambiguous semidirect productC4⋊(C3×D19)

Splittings

FittingC3×C4×C19 ⋊ C2
HallC3×C19 ⋊ D4
SylowC19 ⋊ C3×D4

Subgroups

subgroupG/H
centerC2×C3C2×D19
commutator, cosocleC2×C19C22×C3
FrattiniC2C2×C3×D19
FittingC3×C4×C19C2
socleC2×C3×C19C22
hypercenterC3×C4D19
hypocenterC19C3×D4
LCSG ▷ C2×C19 ▷ C19
UCSC1 ◁ C2×C3 ◁ C3×C4

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 26
transitive degreeμt(G) = 228

A minimal faithful permutation representation — 2 generators on 26 points:

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

Conjugacy classes & character table

123 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744818, #11744819, #11744821, …
maximal normal subgroup of#11744818, #11744819, #11744821, …
maximal quotient ofC3×#64516, C3×D152, C3×Dic76, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 26);