Dic₅×C₆₁ #11927403

order	1220
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₄×C₆₁ · Schur multiplier (H₂) = C₁

Aliases

direct product

Dic₅×C₆₁

Splittings

derived, Sylow	C₅ ⋊ C₄×C₆₁
Hall	C₅×C₆₁ ⋊ C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂×C₆₁	D₅
commutator, hypocenter	C₅	C₄×C₆₁
Frattini	C₂	D₅×C₆₁
Fitting, socle, prime radical	C₂×C₅×C₆₁	C₂
cosocle	C₂×C₅	C₂×C₆₁
LCS	G ▷ C₅
UCS	C₁ ◁ C₂×C₆₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 70
transitive degree	μ_t(G) = 1220

A minimal faithful permutation representation — 2 generators on 70 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70);(1,3,2,4)(6,9)(7,8)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	Dic₅×C₆₁⋊C₃, C₃×Dic₅×C₆₁, Dic₁₅×C₆₁, …
maximal normal subgroup of	Dic₅×C₆₁⋊C₃, C₃×Dic₅×C₆₁, C₁₁×Dic₅×C₆₁, …
maximal quotient of	C₃×Dic₅×C₆₁, Dic₁₅×C₆₁, C₆₁×#4379, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1220, 3);

Dic5×C61 #11927403