C₃₇×#1890 #17990708

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

Aliases

direct product

C₃₇×#1890

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	#120
commutator	C₂³	C₂²×C₄×C₃₇
Frattini, cosocle	C₂⁴	C₂³×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂⁴×C₃₇ ◁ G

386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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)(39,41)(42,43)(44,47)(45,52,53,48)(46,49)(50,54,56,57)(51,55);(38,39,40,41)(42,44,46,51)(43,47,49,55)(45,50,53,56)(48,54,52,57);(42,45)(43,48)(44,50)(46,53)(47,54)(49,52)(51,56)(55,57)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12754, C₃₇×#12760, C₃₇×#12761, …
maximal normal subgroup of	C₃₇×#12754, C₃₇×#12760, C₃₇×#12761, …
maximal quotient of	C₃₇×#8966, C₃₇×#8968, C₃₇×#9011, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 622);