Finite Group Databasebeta

← Prev (#5945) Next (#5947) →

Cayley diagram

C4×C22⋊A4 #5946

order192
isoclinismisoclinic class of C22⋊A4
automorphismAut(G) = C2×AΓL(2,𝔽4) (order 11520) · Inn(G) = C22⋊A4 · Out(G) = C2×S5
homologyAbelianization (H1) = C3×C4 · Schur multiplier (H2) = C24

Aliases

direct productC4×C22⋊A4

Splittings

Fitting, SylowC24×C4 ⋊ C3
derivedC24 ⋊ C3×C4

Subgroups

subgroupG/H
center, hypercenterC4C22⋊A4
commutator, hypocenterC24C3×C4
FrattiniC2C2×C22⋊A4
FittingC24×C4C3
socle, cosocleC25C2×C3
prime radicalC2×C22⋊A4C2
LCSG ▷ C24
UCSC1 ◁ C4

262 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 24

A minimal faithful permutation representation — 3 generators on 12 points:

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

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

derived subgroup of#11711262, #11711263, #11711264, …
maximal subgroup of#1779⋊C3, C4×#120⋊C3, C4⋊(#120⋊C3), …
maximal normal subgroup of#1779⋊C3, C4×#120⋊C3, C4⋊(#120⋊C3), …
maximal quotient ofC4×C22⋊SL(2,𝔽3), C4×#997, C8×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1505);