C₃₇×#1437 #17990255

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₃₇×#1437

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂³
commutator	C₂²	C₂×C₄²×C₃₇
Frattini, cosocle	C₂⁴	C₂³×C₃₇
prime radical	C₂⁵×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ G

828 conjugacy classes of subgroups; only the 280 normal subgroups are shown.

normal subgroups only

This group has 828 conjugacy classes of subgroups — too many to draw. Only its 280 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7213, C₃₇×#8346, C₂×C₃₇×#1437, …
maximal normal subgroup of	C₃₇×#7213, C₃₇×#8346, C₂×C₃₇×#1437, …
maximal quotient of	C₃₇×#7554, C₃₇×#7555, C₃₇×#7556, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 169);