C₃×C₉₇ #64405

order	291
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂×C₃×C₃₂ (order 192) · Inn(G) = C₁ · Out(G) = C₂×C₃×C₃₂
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₂₉₁
abelian	C₃×C₉₇

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) = 100
transitive degree	μ_t(G) = 291

A minimal faithful permutation representation — 1 generator on 100 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100)

Conjugacy classes & character table

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

Relationships

center of	C₃×D₃×C₉₇, H(3,𝔽₃)×C₉₇, 3₋¹⁺²×C₉₇, …
derived subgroup of	D₂₉₁, Dic₂₉₁, C₃⋊#87604, …
Fitting subgroup of	D₃×C₉₇, C₃×D₉₇, D₂₉₁, …
socle of	D₃×C₉₇, C₃×D₉₇, D₂₉₁, …
maximal subgroup of	D₃×C₉₇, C₃×D₉₇, D₂₉₁, …
maximal normal subgroup of	D₃×C₉₇, C₃×D₉₇, D₂₉₁, …
maximal quotient of	C₂×C₃×C₉₇, C₉×C₉₇, C₃²×C₉₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(291, 2);

C3×C97 #64405