Finite Group Databasebeta

← Prev (#5927) Next (#5929) →

Cayley diagram

C22×SL(2,ℤ/4) #5928

order192
isoclinismisoclinic class of S4
automorphismAut(G) = S4×#5934 (order 4608) · Inn(G) = S4 · Out(G) = #5934
homologyAbelianization (H1) = C22×C4 · Schur multiplier (H2) = C24

Aliases

direct productC22×SL(2,ℤ/4)

Splittings

derivedA4 ⋊ C22×C4

Subgroups

subgroupG/H
center, hypercenterC23S4
commutator, hypocenterA4C22×C4
FrattiniC2C22×S4
Fitting, socleC25D3
prime radicalC23×A4C2
cosocleC2×A4C23
LCSG ▷ A4
UCSC1 ◁ C23

277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degreeμ(G) = 12
transitive degreeμt(G) = 48

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

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup ofA4⋊#95, #85292, #85293, …
maximal normal subgroup ofA4⋊#95, #85292, #85293, …
maximal quotient ofC22×A4⋊C8, C2×#5409, C2×C4×SL(2,ℤ/4), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1487);