C₇₁⋊C₇ #92508

order	497
isoclinism	isoclinic class of C₇₁⋊C₇
automorphism	?
homology	Abelianization (H₁) = C₇ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product

C₇₁⋊C₇

Splittings

Fitting, socle, derived, Sylow

C₇₁ ⋊ C₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇₁	C₇
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:

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

Polycyclic presentation:

7,71|b^a=b^45

Conjugacy classes & character table

17 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇₁⋊C₇, C₂×C₇₁⋊C₇, C₃×C₇₁⋊C₇, …
maximal normal subgroup of	D₇₁⋊C₇, C₂×C₇₁⋊C₇, C₃×C₇₁⋊C₇, …
maximal quotient of	C₂×C₇₁⋊C₇, C₃×C₇₁⋊C₇, C₅×C₇₁⋊C₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(497, 1);

C71⋊C7 #92508