C₃₇×#3167 #17991985

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3167

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₃₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 53
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 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)(40,43)(41,42)(44,45)(47,50)(48,52)(51,53);(39,42)(43,45);(46,47,48,50,49,51,52,53);(38,40,41,44)(39,43,42,45)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#9054, C₃₇×#9055, C₃₇×#9717, …
maximal normal subgroup of	C₃₇×#9054, C₃₇×#9055, C₃₇×#9717, …
maximal quotient of	C₃₇×#20816, C₃₇×#21274, C₃₇×#21400, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1899);

C37×#3167 #17991985