Finite Group Databasebeta

Named group families

Standard families of finite groups, each with a few example members.

Cyclic

Examples: C1, C2, C3, C4, C5

Abelian

Examples: C22, C2×C4, C23, C32, C22×C4

AΓL

Examples: AΓL(1,𝔽4), AΓL(1,𝔽9), AΓL(1,𝔽8), AΓL(1,𝔽16), AΓL(1,𝔽25)

AGL

Examples: AGL(1,𝔽2), AGL(1,𝔽3), AGL(1,ℤ/4), AGL(1,𝔽4), AGL(1,𝔽5)

Alternating

Examples: A2, A3, A4, A5, A6

AΣL

Examples: AΣL(1,𝔽4), AΣL(1,𝔽9), AΣL(1,𝔽8), AΣL(1,𝔽25), AΣL(1,𝔽16)

Borel

Examples: B(2,𝔽2), B(3,𝔽2), B(2,𝔽3), B(2,ℤ/4), B(2,𝔽4)

Dicyclic

Examples: Dic2, Dic3, Dic4, Dic5, Dic6

Dihedral

Examples: D3, D4, D5, D6, D7

Extraspecial

Examples: 2+1+2, 2−1+2, 3+1+2, 3−1+2, 2+1+4

ΓL

Examples: ΓL(1,𝔽4), ΓL(1,𝔽9), ΓL(1,𝔽8), ΓL(1,𝔽25), ΓL(1,𝔽16)

GL

Examples: GL(2,𝔽2), GL(2,𝔽3), GL(2,ℤ/4), GL(3,𝔽2), GL(2,𝔽4)

GO

Examples: GO(3,𝔽3), GO+(4,𝔽2), GO−(4,𝔽2), GO(3,𝔽5), GO(3,𝔽7)

GU

Examples: GU(2,𝔽2), GU(2,𝔽3), GU(2,𝔽4), GU(3,𝔽2), GU(2,𝔽5)

H

Examples: H(3,𝔽2), H(3,𝔽3), H(5,𝔽2), H(3,ℤ/4), H(3,𝔽4)

Modular maximal-cyclic

Examples: M16, M32, M64, M128, M256

Ω

Examples: Ω(3,𝔽3), Ω+(4,𝔽2), Ω−(4,𝔽2), Ω(3,𝔽5), Ω(3,𝔽7)

PΓL

Examples: PΓL(2,𝔽4), PΓL(2,𝔽9), PΓL(2,𝔽8), PΓL(2,𝔽16), PΓL(2,𝔽25)

PGL

Examples: PGL(2,𝔽2), PGL(2,𝔽3), PGL(2,𝔽4), PGL(2,𝔽5), PGL(3,𝔽2)

PΣL

Examples: PΣL(2,𝔽4), PΣL(2,𝔽9), PΣL(2,𝔽8), PΣL(2,𝔽25), PΣL(2,𝔽16)

PSL

Examples: PSL(2,𝔽2), PSL(2,𝔽3), PSL(2,𝔽4), PSL(2,𝔽5), PSL(2,𝔽7)

PSp

Examples: PSp(2,𝔽2), PSp(2,𝔽3), PSp(2,𝔽5), PSp(2,𝔽4), PSp(2,𝔽7)

PSU

Examples: PSU(2,𝔽2), PSU(2,𝔽3), PSU(2,𝔽4), PSU(2,𝔽5), PSU(3,𝔽2)

Quaternion

Examples: Q8, Q16, Q32, Q64, Q128

Semi-dihedral

Examples: SD16, SD32, SD64, SD128, SD256

ΣL

Examples: ΣL(2,𝔽4), ΣL(2,𝔽9), ΣL(2,𝔽8), ΣL(2,𝔽16), ΣL(2,𝔽25)

SL

Examples: SL(2,𝔽2), SL(2,𝔽3), SL(2,ℤ/4), SL(2,𝔽4), SL(2,𝔽5)

SO

Examples: SO(3,𝔽3), SO+(4,𝔽2), SO−(4,𝔽2), SO(3,𝔽5), SO(3,𝔽7)

Sp

Examples: Sp(2,𝔽2), Sp(2,𝔽3), Sp(2,ℤ/4), Sp(2,𝔽4), Sp(2,𝔽5)

Sporadic

Examples: M11

SU

Examples: SU(2,𝔽2), SU(2,𝔽3), SU(2,𝔽4), SU(2,𝔽5), SU(3,𝔽2)

Symmetric

Examples: S1, S2, S3, S4, S5

UT

Examples: UT(3,𝔽2), UT(3,𝔽3), UT(3,ℤ/4), UT(4,𝔽2), UT(3,𝔽4)