C₂²×C₇×C₄₉ #13063142

order	1372
nilpotency class	1
isoclinism	isoclinic class of C₁
automorphism	Aut(G) order 74088
homology	Abelianization (H₁) = G · Schur multiplier (H₂) = C₂×C₇

Aliases

abelian

C₂²×C₇×C₄₉

Subgroups

	subgroup	G/H
center	G	C₁
commutator	C₁	G
Frattini, cosocle	C₇	C₂²×C₇²
socle, prime radical	C₂²×C₇²	C₇
LCS	G ▷ C₁
UCS	C₁ ◁ G

90 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)(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);(3,4)(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	C₂×C₄×C₇×C₄₉, C₂³×C₇×C₄₉, C₂²×C₃×C₇×C₄₉, …
maximal normal subgroup of	C₂×C₄×C₇×C₄₉, C₂³×C₇×C₄₉, C₂²×C₃×C₇×C₄₉, …
maximal quotient of	C₂×C₄×C₇×C₄₉, C₂³×C₇×C₄₉, C₂²×C₃×C₇×C₄₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1372, 23);

C22×C7×C49 #13063142