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Cayley diagram

C2×C22⋊(C22⋊A4) #87589

order384
isoclinismisoclinic class of C22⋊(C22⋊A4)
automorphism?
homologyAbelianization (H1) = C2×C3 · Schur multiplier (H2) = C29

Aliases

direct productC2×C22⋊(C22⋊A4)

Splittings

Fitting, socle, SylowC27 ⋊ C3
derivedC26 ⋊ C2×C3

Subgroups

subgroupG/H
center, hypercenterC2C22⋊(C22⋊A4)
commutator, cosocle, hypocenterC26C2×C3
FrattiniC1G
Fitting, socleC27C3
LCSG ▷ C26
UCSC1 ◁ C2

9884 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 14
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 4 generators on 14 points:

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

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

derived subgroup of#72233209, #72233210, #72233211, …
maximal subgroup of#11709759, C2×#85645, C4×C22⋊(C22⋊A4), …
maximal normal subgroup of#11709759, C2×#85645, C4×C22⋊(C22⋊A4), …
maximal quotient ofC4×C22⋊(C22⋊A4), C2×C22⋊(C22⋊SL(2,𝔽3)), #11710913, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 20165);