C₃×C₇×Dic₃₁ #14579607

order	2604
isoclinism	isoclinic class of D₃₁
automorphism	?
homology	Abelianization (H₁) = C₃×C₄×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₇×Dic₃₁

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₃×C₇	D₃₁
commutator, hypocenter	C₃₁	C₃×C₄×C₇
Frattini	C₂	C₃×C₇×D₃₁
Fitting, socle, prime radical	C₂×C₃×C₇×C₃₁	C₂
cosocle	C₂×C₃₁	C₂×C₃×C₇
LCS	G ▷ C₃₁
UCS	C₁ ◁ C₂×C₃×C₇

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);(11,13,12,14)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₅×C₇×Dic₃₁, C₃×C₇×Dic₁₅₅, C₃×C₇×Dic₃₁⋊C₅, …
maximal normal subgroup of	C₃×C₅×C₇×Dic₃₁, C₃×C₇×Dic₃₁⋊C₅, C₃×C₄₉×Dic₃₁, …
maximal quotient of	C₃×C₅×C₇×Dic₃₁, C₃×C₇×Dic₁₅₅, C₃×C₇×C₃₁⋊AGL(1,𝔽₅), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2604, 18);

C3×C7×Dic31 #14579607