Finite Group Databasebeta

← Prev (#247) Next (#249) →

Cayley diagram

C22⋊A4 #248

order48
isoclinismisoclinic class of C22⋊A4
automorphismAut(G) = AΓL(2,𝔽4) (order 5760) · Inn(G) = G · Out(G) = S5
homologyAbelianization (H1) = C3 · Schur multiplier (H2) = C24

Aliases

unambiguous semidirect productC22⋊A4

Splittings

Fitting, socle, derived, SylowC24 ⋊ C3

Subgroups

subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, cosocle, hypocenterC24C3
LCSG ▷ C24
UCSC1

34 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 8
transitive degreeμt(G) = 12

A minimal faithful permutation representation — 3 generators on 8 points:

(2,3,4)(6,7,8);(5,8)(6,7);(1,4)(2,3)

Polycyclic presentation:

3,2^4|b^a=c,c^a=bc,d^a=e,e^a=de

Conjugacy classes & character table

C1C2C3C4C5C6C7C8
size1333331616
order12222233
χ111111111
χ2111111−1+i32−1−i32
χ3111111−1−i32−1+i32
χ433-1-1-1-100
χ53-13-1-1-100
χ63-1-13-1-100
χ73-1-1-13-100
χ83-1-1-1-1300

Here i is the imaginary unit (i² = −1).

Relationships

derived subgroup of#1020, #4625, #1020⋊C2, …
maximal subgroup of#120⋊C3, #1020, C2×C22⋊A4, …
maximal normal subgroup of#120⋊C3, #1020, C2×C22⋊A4, …
maximal quotient ofC22⋊SL(2,𝔽3), #997, C2×C22⋊A4, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 50);