C₃₇×#1849 #17990667

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

Aliases

direct product

C₃₇×#1849

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×C₄⋊C₄
prime radical	C₂³×C₃₇	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 57 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)(42,43,46,49,47,50,45,55)(44,51,54,56,52,57,53,48);(42,44,45,53,47,52,46,54)(43,48,49,51,50,56,55,57);(38,39,40,41)(44,52)(48,56)(51,57)(53,54)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7374, C₃₇×#8916, C₃₇×#10524, …
maximal normal subgroup of	C₃₇×#7374, C₃₇×#8916, C₃₇×#10524, …
maximal quotient of	C₃₇×#8808, C₃₇×#8809, C₃₇×#8810, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 581);

C37×#1849 #17990667