C₃₇×#2296 #17991114

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#2296

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂³
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
socle	C₂³×C₃₇	C₂⁴
prime radical	C₂⁴×C₃₇	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ G

656 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)(40,41)(42,43,45,44)(46,47)(48,51)(49,50)(52,53);(39,41)(42,44,45,43);(47,50)(51,53);(46,48,49,52)(47,51,50,53)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7564, C₃₇×#7579, C₃₇×#7598, …
maximal normal subgroup of	C₃₇×#7564, C₃₇×#7579, C₃₇×#7598, …
maximal quotient of	C₃₇×#13756, C₃₇×#13759, C₄×C₃₇×#385, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1028);

C37×#2296 #17991114