order	248
isoclinism	isoclinic class of D₃₁
automorphism	Aut(G) = C₂²×AGL(1,𝔽₃₁) (order 3720) · Inn(G) = D₃₁ · Out(G) = C₂²×C₃×C₅
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₄×D₃₁

Splittings

Fitting	C₄×C₃₁ ⋊ C₂
derived, Sylow	C₃₁ ⋊ C₂×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₄	D₃₁
commutator, hypocenter	C₃₁	C₂×C₄
Frattini	C₂	C₂×D₃₁
Fitting	C₄×C₃₁	C₂
socle, cosocle	C₂×C₃₁	C₂²
prime radical	C₂×D₃₁	C₂
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 35
transitive degree	μ_t(G) = 124

A minimal faithful permutation representation — 2 generators on 35 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(20,21)(22,23)(24,25)(26,27)(28,29)(30,31);(1,2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,31,29,27,25,23,21,19,17,15,13,11,9,7,5,3)(32,33,34,35)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₃₁, #92469, C₂×C₄×D₃₁, …
maximal normal subgroup of	C₈×D₃₁, #92469, C₂×C₄×D₃₁, …
maximal quotient of	C₈×D₃₁, #92469, C₄×Dic₃₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 4);

C4×D31 #6936

Aliases

Splittings

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₄×D₃₁ #6936