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Cayley diagram

C22×C24⋊C5 #66299

order320
isoclinismisoclinic class of C24⋊C5
automorphism?
homologyAbelianization (H1) = C22×C5 · Schur multiplier (H2) = C23

Aliases

direct productC22×C24⋊C5
unambiguous semidirect productC26⋊C5

Splittings

Fitting, socle, SylowC26 ⋊ C5
derivedC24 ⋊ C22×C5

Subgroups

subgroupG/H
center, hypercenterC22C24⋊C5
commutator, cosocle, hypocenterC24C22×C5
FrattiniC1G
Fitting, socleC26C5
LCSG ▷ C24
UCSC1 ◁ C22

583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 20

A minimal faithful permutation representation — 2 generators on 12 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of((C24⋊C5)⋊C2)⋊A4, A4×(C24⋊C5)⋊C2
maximal subgroup ofC2×#66245, C22⋊((C24⋊C5)⋊C2), C2×C4×C24⋊C5, …
maximal normal subgroup ofC2×#66245, C22⋊((C24⋊C5)⋊C2), C2×C4×C24⋊C5, …
maximal quotient ofC2×C4×C24⋊C5, D4×C24⋊C5, Q8×C24⋊C5, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(320, 1637);