Finite Group Databasebeta

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Cayley diagram

C22×D57 #90733

order456
isoclinismisoclinic class of D57
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C23

Aliases

direct productC22×D57

Splittings

Fitting, socleC22×C3×C19 ⋊ C2
derived, HallC3×C19 ⋊ C23
SylowC19 ⋊ C22×D3
SylowC3 ⋊ C22×D19

Subgroups

subgroupG/H
center, hypercenterC22D57
commutator, cosocle, hypocenterC3×C19C23
FrattiniC1G
Fitting, socleC22×C3×C19C2
LCSG ▷ C3×C19
UCSC1 ◁ C22

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 26 points:

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

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11744833, #11744889, C2×D57⋊C4, …
maximal normal subgroup of#11744833, #11744889, C2×D57⋊C4, …
maximal quotient ofC2×Dic114, C2×C4×D57, C2×D228, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 53);