C₃₇×#1806 #17990624

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1806

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂×#41×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 49 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)(39,41)(42,43,45,48,46,44,49,47);(38,39,40,41);(42,44)(43,46)(45,47)(48,49)

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8420, C₃₇×#8421, C₃₇×#8422, …
maximal normal subgroup of	C₃₇×#8420, C₃₇×#8421, C₃₇×#8422, …
maximal quotient of	C₃₇×#8563, C₃₇×#8564, C₃₇×#8565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 538);

C37×#1806 #17990624