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Cayley diagram

C7×C22⋊A4 #66754

order336
isoclinismisoclinic class of C22⋊A4
automorphism?
homologyAbelianization (H1) = C3×C7 · Schur multiplier (H2) = C24

Aliases

direct productC7×C22⋊A4
unambiguous semidirect productC22⋊(C7×A4)

Splittings

Fitting, socle, HallC24×C7 ⋊ C3
derived, SylowC24 ⋊ C3×C7

Subgroups

subgroupG/H
center, hypercenterC7C22⋊A4
commutator, cosocle, hypocenterC24C3×C7
FrattiniC1G
Fitting, socleC24×C7C3
LCSG ▷ C24
UCSC1 ◁ C7

68 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 15
transitive degreeμt(G) = 84

A minimal faithful permutation representation — 3 generators on 15 points:

(2,3,4)(6,7,8)(9,10,11,12,13,14,15);(5,8)(6,7);(1,4)(2,3)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

derived subgroup ofC7⋊#1020, C7⋊#4625, #1020⋊D7, …
maximal subgroup of#10620796, C7×#120⋊C3, C7×#1020, …
maximal normal subgroup of#10620796, C7×#120⋊C3, C7×#1020, …
maximal quotient ofC7×C22⋊SL(2,𝔽3), C7×#997, C2×C7×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(336, 223);