C₃₁×C₃⋊D₁₅ #14584087

order	2790
isoclinism	isoclinic class of C₃⋊D₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃₁ · Schur multiplier (H₂) = C₃

Aliases

direct product	C₃₁×C₃⋊D₁₅
unambiguous semidirect product	C₃⋊(D₁₅×C₃₁)
unambiguous semidirect product	C₅⋊(C₃⋊D₃×C₃₁)

Splittings

Fitting, socle, Hall	C₃²×C₅×C₃₁ ⋊ C₂
derived, Hall	C₃²×C₅ ⋊ C₂×C₃₁
Hall	C₃²×C₃₁ ⋊ D₅
Hall	C₅×C₃₁ ⋊ C₃⋊D₃
Sylow	C₃² ⋊ D₅×C₃₁
Sylow	C₅ ⋊ C₃⋊D₃×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃₁	C₃⋊D₁₅
commutator, cosocle, hypocenter	C₃²×C₅	C₂×C₃₁
Frattini	C₁	G
Fitting, socle	C₃²×C₅×C₃₁	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₃₁

48 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 42
transitive degree	μ_t(G) = 1395

A minimal faithful permutation representation — 3 generators on 42 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31)(33,34)(35,36)(38,39)(41,42);(32,33,35,36,34)(40,41,42);(37,38,39)

Conjugacy classes & character table

744 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₁×C₃⋊D₇₅, C₃⋊D₁₅×ΓL(1,𝔽₃₂), C₃₁×C₃⋊(C₃⋊(C₅⋊D₅)), …
maximal normal subgroup of	C₃⋊D₁₅×ΓL(1,𝔽₃₂), C₅×C₃₁×C₃⋊D₁₅, C₇×C₃₁×C₃⋊D₁₅, …
maximal quotient of	C₃₁×C₃⋊D₇₅, C₃₁×C₃⋊(C₃⋊(C₅⋊D₅)), C₅×C₃₁×C₃⋊D₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2790, 47);

C31×C3⋊D15 #14584087