Finite Group Databasebeta

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Cayley diagram

C7⋊#401 #89434

order448
isoclinismisoclinic class of #89428
automorphism?
homologyAbelianization (H1) = C23 · Schur multiplier (H2) = C22

Aliases

unambiguous semidirect productC7⋊#401

Splittings

SylowC7 ⋊ #401

Subgroups

subgroupG/H
center, FrattiniC23C22×D7
commutator, socle, prime radical, cosocleC23×C7C23
FittingC7×#95C2
hypercenter#95D7
hypocenterC7#401
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #95

118 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 31
transitive degreeμt(G) = 448

A minimal faithful permutation representation — 3 generators on 31 points:

(1,2,4,6,7,5,3)(9,13)(12,15)(18,23)(20,22)(24,26,27,31)(25,28,29,30);(8,10,11,14)(9,12,13,15)(16,18)(17,20)(19,23)(21,22)(26,31)(28,30);(2,3)(4,5)(6,7)(8,9)(10,12)(11,13)(14,15)(16,17,19,21)(18,22,23,20)(24,25,27,29)(26,30,31,28)

Polycyclic presentation:

2^6,7|a^2=f,b^2=d,c^2=ef,b^a=be,c^a=cdef,g^a=g^6,c^b=cf

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofC2×C7⋊#401, #11730798, #11730801, …
maximal normal subgroup ofC2×C7⋊#401, #11730798, #11730801, …
maximal quotient ofC7⋊#2104, #11727170, #11727171, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(448, 196);