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Cayley diagram

C7×Dic14 #87643

order392
isoclinismisoclinic class of Dic14
automorphism?
homologyAbelianization (H1) = C22×C7 · Schur multiplier (H2) = C1

Aliases

direct productC7×Dic14
unambiguous semidirect productC7⋊(C7×Q8)

Splittings

SylowC72 ⋊ Q8

Subgroups

subgroupG/H
centerC2×C7C2×D7
commutator, cosocleC2×C7C22×C7
FrattiniC2C2×C7×D7
FittingC4×C72C2
socle, prime radicalC2×C72C22
hypercenterC4×C7D7
hypocenterC7C7×Q8
LCSG ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C7 ◁ C4×C7

33 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 22
transitive degreeμt(G) = 56

A minimal faithful permutation representation — 2 generators on 22 points:

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

Conjugacy classes & character table

119 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#11716034, #11716036, #11716037, …
maximal normal subgroup of#11716034, #11716036, #11716037, …
maximal quotient ofC7×Dic7⋊C4, C7×C4⋊Dic7, C2×C7×Dic14, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(392, 23);