C₇₁⋊C₅ #67059

order	355
isoclinism	isoclinic class of C₇₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₅ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₇₁⋊C₅

Splittings

Fitting, socle, derived, Sylow

C₇₁ ⋊ C₅

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₁	C₅
LCS	G ▷ C₇₁
UCS	C₁

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 71
transitive degree	μ_t(G) = 71

A minimal faithful permutation representation — 2 generators on 71 points:

(2,3,5,9,17)(4,7,13,24,39)(6,11,21,25,30)(8,15,28,45,16)(10,19,32,18,31)(12,23,38,57,58)(14,26,42,60,49)(20,34,53,40,59)(22,36,46,41,50)(27,44,63,48,67)(29,47,65,37,56)(33,52,35,43,51)(54,70,61,71,68)(55,66,64,62,69);(1,2,4,8,16,5,10,20,31,50,13,25,41,52,69,28,46,64,70,71,15,29,48,34,54,17,30,49,56,45,32,51,68,59,65,53,60,7,14,27,19,33,23,24,40,36,55,44,42,61,39,58,67,63,3,6,12,9,18,11,22,37,21,35,26,43,62,38,57,47,66)

Polycyclic presentation:

5,71|b^a=b^54

Conjugacy classes & character table

19 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇₁⋊C₅, C₂×C₇₁⋊C₅, C₃×C₇₁⋊C₅, …
maximal normal subgroup of	D₇₁⋊C₅, C₂×C₇₁⋊C₅, C₃×C₇₁⋊C₅, …
maximal quotient of	C₂×C₇₁⋊C₅, C₃×C₇₁⋊C₅, C₇₁⋊C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(355, 1);

C71⋊C5 #67059