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Cayley diagram

GL(2,ℤ/4) #988

order96
automorphismAut(G) = C22×S4 (order 96) · Inn(G) = C2×S4 · Out(G) = C2
homologyAbelianization (H1) = C22 · Schur multiplier (H2) = C22

Aliases

general linearGL(2,ℤ/4)

Splittings

FittingC24 ⋊ D3

Subgroups

subgroupG/H
center, FrattiniC2C2×S4
commutator, cosocleC2×A4C22
FittingC24D3
socleC23C2×D3
hypercenterC22S4
hypocenterA4D4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C2 ◁ C22

62 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 8 points:

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

Polycyclic presentation:

2^3,3,2^2|b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=f,f^d=ef

Conjugacy classes & character table

14 conjugacy classes — view the full character table.

Relationships

maximal subgroup of#5912, D4×S4, #5914, …
maximal normal subgroup of#5912, D4×S4, #5914, …
maximal quotient ofSL(2,ℤ/4)⋊C4, (C2×S4)⋊C4, #5414, …
automorphism group ofC42
outer automorphism group ofC42, #95, #338, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(96, 195);