C₇×Dic₆₀ #13132341

order	1680
isoclinism	isoclinic class of #6706
automorphism	?
homology	Abelianization (H₁) = C₂²×C₇ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₇×Dic₆₀

Splittings

Hall	C₃×C₅×C₇ ⋊ Q₁₆
Hall	C₅×C₇ ⋊ Dic₁₂
Hall	C₃×C₇ ⋊ Dic₂₀
Hall	C₃×C₅ ⋊ C₇×Q₁₆
Sylow	C₅ ⋊ C₇×Dic₁₂
Sylow	C₃ ⋊ C₇×Dic₂₀

Subgroups

	subgroup	G/H
center	C₂×C₇	D₆₀
commutator, cosocle	C₃×C₄×C₅	C₂²×C₇
Frattini	C₄	C₂×C₇×D₁₅
Fitting	C₃×C₅×C₇×C₈	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₄
hypercenter	C₇×C₈	D₁₅
hypocenter	C₃×C₅	C₇×Q₁₆
LCS	G ▷ C₃×C₄×C₅ ▷ C₂×C₃×C₅ ▷ C₃×C₅
UCS	C₁ ◁ C₂×C₇ ◁ C₄×C₇ ◁ C₇×C₈

72 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 31
transitive degree	μ_t(G) = 1680

A minimal faithful permutation representation — 2 generators on 31 points:

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

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 361);

C7×Dic60 #13132341