C₃₇×#1848 #17990666

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

Aliases

direct product

C₃₇×#1848

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

220 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)(44,51,53,56,54,57,52,48);(42,44,45,52,47,54,46,53)(43,48,49,51,50,56,55,57);(38,39,40,41)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7270, C₃₇×#7272, C₃₇×#7273, …
maximal normal subgroup of	C₃₇×#7270, C₃₇×#7272, C₃₇×#7273, …
maximal quotient of	C₃₇×#8805, C₃₇×#8806, C₃₇×#8807, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 580);

C37×#1848 #17990666