Finite Group Databasebeta

← Prev (#64595) Next (#64597) →

(C₂²⋊D₁₃)⋊C₃ #64596

order	312
isoclinism	isoclinic class of (C₁₃⋊C₃)⋊Q₈
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₃) (order 624) · Inn(G) = C₂×D₁₃⋊C₃ · Out(G) = C₄
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product

(C₂²⋊D₁₃)⋊C₃

Splittings

Fitting	C₂²×C₁₃ ⋊ C₂×C₃
Hall	C₂²⋊D₁₃ ⋊ C₃
Hall	C₁₃⋊C₃ ⋊ D₄
Sylow	C₁₃ ⋊ C₃×D₄

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃⋊C₃
commutator, socle, cosocle	C₂×C₁₃	C₂²×C₃
Fitting	C₂²×C₁₃	C₂×C₃
hypercenter	C₂²	D₁₃⋊C₃
hypocenter	C₁₃	C₃×D₄
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂ ◁ C₂²

32 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

(1,2,8)(3,9,13)(4,6,11)(5,12,7)(15,17);(2,3)(4,6)(5,9)(7,8)(10,12)(11,13)(14,15)(16,17)

Polycyclic presentation:

2^2,3,2,13|b^a=bd,e^a=e^12,e^c=e^3

Conjugacy classes & character table

23 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6127⋊C₃, D₄×D₁₃⋊C₃, #6129⋊C₃, …
maximal normal subgroup of	#6127⋊C₃, D₄×D₁₃⋊C₃, #6129⋊C₃, …
maximal quotient of	#6101⋊C₃, #6103⋊C₃, #6104⋊C₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 12);