C₃₇×#1904 #17990722

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#1904

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	#386
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂×AGL(1,ℤ/8)×C₃₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

346 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 592

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

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×C₃₇×#1904, C₃₇×#20998, C₃₇×#22284, …
maximal normal subgroup of	C₂×C₃₇×#1904, C₃₇×#20998, C₃₇×#22284, …
maximal quotient of	C₃₇×#9087, C₃₇×#9089, C₃₇×#9091, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 636);