C₆₁⋊C₅ #64554

order	305
isoclinism	isoclinic class of C₆₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₆₁⋊C₅

Splittings

Fitting, socle, derived, Sylow

C₆₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₆₁	C₅
LCS	G ▷ C₆₁
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 61

A minimal faithful permutation representation — 2 generators on 61 points:

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

Polycyclic presentation:

5,61|b^a=b^9

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₆₁⋊C₅, C₂×C₆₁⋊C₅, (C₆₁⋊C₃)⋊C₅, …
maximal normal subgroup of	D₆₁⋊C₅, C₂×C₆₁⋊C₅, (C₆₁⋊C₃)⋊C₅, …
maximal quotient of	C₂×C₆₁⋊C₅, C₃×C₆₁⋊C₅, C₆₁⋊C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(305, 1);

C61⋊C5 #64554