C₃₇×#1933 #17990751

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1933

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

256 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 53 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)(42,43,45,44)(47,51)(48,53)(49,52);(42,44,45,43)(46,47,49,51,50,48,52,53);(39,41)(46,48,49,53,50,47,52,51)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#1933, C₃₇×#21083, C₃₇×#21087, …
maximal normal subgroup of	C₂×C₃₇×#1933, C₃₇×#21083, C₃₇×#21087, …
maximal quotient of	C₃₇×#9553, C₃₇×#9557, C₃₇×#9561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 665);

C37×#1933 #17990751