Finite Group Databasebeta

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Cayley diagram

D7×#157 #92740

order504
isoclinismisoclinic class of D7×#157
automorphism?
homologyAbelianization (H1) = C2×C4 · Schur multiplier (H2) = C2×C3

Aliases

direct productD7×#157

Splittings

Fitting, socle, derived, HallC32×C7 ⋊ C2×C4
SylowC32 ⋊ C4×D7
SylowC7 ⋊ C2×#157

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C7C2×C4
prime radicalD7×C3⋊D3C2
cosocleC7×C3⋊D3C22
LCSG ▷ C32×C7
UCSC1

52 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 13
transitive degreeμt(G) = 42

A minimal faithful permutation representation — 2 generators on 13 points:

(1,2)(3,5,6,4)(7,8,10,12,13,11,9);(1,3,6)(2,5,4)(8,9)(10,11)(12,13)

Conjugacy classes & character table

30 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofD7×AGL(1,𝔽9), D7×GO+(4,𝔽2), D7×PSU(3,𝔽2), …
maximal normal subgroup ofD7×AGL(1,𝔽9), D7×GO+(4,𝔽2), D7×PSU(3,𝔽2), …
maximal quotient ofDic7×#157, (C2×D7)⋊#157, Dic7⋊#157, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(504, 166);