C₃₁⋊C₈ #6933

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) = C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₃₁⋊C₈

Splittings

derived, Sylow

C₃₁ ⋊ C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₃₁
commutator, hypocenter	C₃₁	C₈
Fitting, cosocle	C₄×C₃₁	C₂
socle, prime radical	C₂×C₃₁	C₄
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₄

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 39
transitive degree	μ_t(G) = 248

A minimal faithful permutation representation — 2 generators on 39 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)(32,33,34,36,35,37,38,39);(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)

Polycyclic presentation:

2^3,31|a^2=b,b^2=c,d^a=d^30

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₈×D₃₁, #92469, C₂×C₃₁⋊C₈, …
maximal normal subgroup of	C₈×D₃₁, #92469, C₂×C₃₁⋊C₈, …
maximal quotient of	C₃₁⋊C₁₆, C₂×C₃₁⋊C₈, C₃×C₃₁⋊C₈, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(248, 1);

C31⋊C8 #6933