C₃×Dic₇×C₃₁ #14579608

order	2604
isoclinism	isoclinic class of D₇
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₃₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×Dic₇×C₃₁

Splittings

derived, Sylow	C₇ ⋊ C₃×C₄×C₃₁
Hall	C₃×C₇×C₃₁ ⋊ C₄
Hall	C₇×C₃₁ ⋊ C₃×C₄
Hall	C₃×C₇ ⋊ C₄×C₃₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₃₁	D₇
commutator, hypocenter	C₇	C₃×C₄×C₃₁
Frattini	C₂	C₃×D₇×C₃₁
Fitting, socle, prime radical	C₂×C₃×C₇×C₃₁	C₂
cosocle	C₂×C₇	C₂×C₃×C₃₁
LCS	G ▷ C₇
UCS	C₁ ◁ C₂×C₃×C₃₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 45
transitive degree	μ_t(G) = 2604

A minimal faithful permutation representation — 2 generators on 45 points:

(1,2,3)(4,5)(6,7)(8,9,10,11,12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45);(4,6,5,7)(9,14)(10,13)(11,12)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₅×Dic₇×C₃₁, C₃×C₃₁×Dic₃₅, C₃×Dic₇×ΓL(1,𝔽₃₂), …
maximal normal subgroup of	C₃×C₅×Dic₇×C₃₁, C₃×Dic₇×ΓL(1,𝔽₃₂), C₃×C₇×Dic₇×C₃₁, …
maximal quotient of	C₃×C₅×Dic₇×C₃₁, C₃×C₃₁×Dic₃₅, C₃×C₃₁×C₇⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2604, 19);

C3×Dic7×C31 #14579608