C₄×C₁₂₁ #92144

order	484
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) = C₂²×C₅×C₁₁ (order 220) · Inn(G) = C₁ · Out(G) = C₂²×C₅×C₁₁
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₁

Aliases

cyclic	C₄₈₄
abelian	C₄×C₁₂₁

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, socle, prime radical, cosocle	C₂×C₁₁	C₂×C₁₁
LCS	G ▷ C₁
UCS	C₁ ◁ G

9 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 484

A minimal faithful permutation representation — 1 generator on 125 points:

(1,2,3,4)(5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125)

Conjugacy classes & character table

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

Relationships

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(484, 2);

center of	M₁₆×C₁₂₁, #41×C₁₂₁, C₄×C₁₃⋊C₃×C₁₂₁, …
derived subgroup of	Dic₄₈₄, #14515412, D₉₆₈, …
Frattini subgroup of	Dic₁₅₉₇₂, D₃₁₉₄₄
Fitting subgroup of	C₁₂₁⋊C₈, Dic₂₄₂, C₄×D₁₂₁, …
maximal subgroup of	C₁₂₁⋊C₈, C₈×C₁₂₁, Dic₂₄₂, …
maximal normal subgroup of	C₁₂₁⋊C₈, C₈×C₁₂₁, Dic₂₄₂, …
maximal quotient of	C₈×C₁₂₁, C₂×C₄×C₁₂₁, Dic₃×C₁₂₁, …

C4×C121 #92144