C₃₇×#2003 #17990821

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#2003

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
prime radical	C₂×AGL(1,ℤ/8)×C₃₇	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

382 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 57
transitive degree	μ_t(G) = 1184

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×C₃₇×#2003, C₃₇×#23399, C₃₇×#23400, …
maximal normal subgroup of	C₂×C₃₇×#2003, C₃₇×#23399, C₃₇×#23400, …
maximal quotient of	C₃₇×#9915, C₃₇×#9921, C₃₇×#9923, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 735);