C₇×C₆₁ #88342

order	427
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃²×C₄×C₅ (order 360) · Inn(G) = C₁ · Out(G) = C₂×C₃²×C₄×C₅
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₂₇
abelian	C₇×C₆₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator, Frattini	C₁	G
LCS	G ▷ C₁
UCS	C₁ ◁ G

4 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 427

A minimal faithful permutation representation — 1 generator on 68 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)

Conjugacy classes & character table

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

Relationships

center of	D₃×C₇×C₆₁, C₇×D₅×C₆₁, C₇×D₁₇×C₆₁, …
derived subgroup of	D₄₂₇, #13046196, #13046197, …
Fitting subgroup of	D₇×C₆₁, C₇×D₆₁, D₄₂₇, …
socle of	D₇×C₆₁, C₇×D₆₁, D₄₂₇, …
maximal subgroup of	D₇×C₆₁, C₇×D₆₁, D₄₂₇, …
maximal normal subgroup of	D₇×C₆₁, C₇×D₆₁, D₄₂₇, …
maximal quotient of	C₂×C₇×C₆₁, C₃×C₇×C₆₁, C₅×C₇×C₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(427, 1);

C7×C61 #88342