C₃₇×#1762 #17990580

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₇×#1762

Subgroups

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

388 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 168 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)(38,39)(40,41)(42,43,45,44)(47,51)(48,49)(52,53);(39,41)(46,47,49,53,50,52,48,51);(42,44,45,43)(47,52)(51,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8416, C₃₇×#8418, C₃₇×#9163, …
maximal normal subgroup of	C₃₇×#8416, C₃₇×#8418, C₃₇×#9163, …
maximal quotient of	C₃₇×#8170, C₃₇×#8172, C₃₇×#8179, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 494);