C₄×C₇×C₄₉ #13063130

order	1372
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 24696
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₇

Aliases

abelian

C₄×C₇×C₄₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₂×C₇	C₂×C₇²
socle, prime radical	C₂×C₇²	C₂×C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

54 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 60
transitive degree	μ_t(G) = 1372

A minimal faithful permutation representation — 2 generators on 60 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)

Conjugacy classes & character table

1372 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₇×C₈×C₄₉, C₂×C₄×C₇×C₄₉, C₃×C₄×C₇×C₄₉, …
maximal normal subgroup of	C₇×C₈×C₄₉, C₂×C₄×C₇×C₄₉, C₃×C₄×C₇×C₄₉, …
maximal quotient of	C₇×C₈×C₄₉, C₂×C₄×C₇×C₄₉, C₃×C₄×C₇×C₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1372, 11);

C4×C7×C49 #13063130