C₅×Dic₈₄ #13132325

order	1680
isoclinism	isoclinic class of #66623
automorphism	?
homology	Abelianization (H₁) = C₂²×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₅×Dic₈₄

Splittings

Hall	C₃×C₅×C₇ ⋊ Q₁₆
Hall	C₅×C₇ ⋊ Dic₁₂
Hall	C₃×C₇ ⋊ C₅×Q₁₆
Hall	C₃×C₅ ⋊ Dic₂₈
Sylow	C₇ ⋊ C₅×Dic₁₂
Sylow	C₃ ⋊ C₅×Dic₂₈

Subgroups

	subgroup	G/H
center	C₂×C₅	D₈₄
commutator, cosocle	C₃×C₄×C₇	C₂²×C₅
Frattini	C₄	C₂×C₅×D₂₁
Fitting	C₃×C₅×C₇×C₈	C₂
socle, prime radical	C₂×C₃×C₅×C₇	D₄
hypercenter	C₅×C₈	D₂₁
hypocenter	C₃×C₇	C₅×Q₁₆
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂×C₅ ◁ C₄×C₅ ◁ C₅×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)(14,21,20,19,18,17,16,15)(22,23,24)(25,26,27,28,29,30,31);(6,14,10,18)(7,15,11,19)(8,16,12,20)(9,17,13,21)(23,24)(26,31)(27,30)(28,29)

Conjugacy classes & character table

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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1680, 345);

C5×Dic84 #13132325