C₃₇×#1800 #17990618

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1800

Subgroups

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

336 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 49 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)(42,43,44,47,45,46,48,49);(38,39,40,41);(43,46)(44,48)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7236, C₃₇×#7321, C₃₇×#8353, …
maximal normal subgroup of	C₃₇×#7236, C₃₇×#7321, C₃₇×#8353, …
maximal quotient of	C₃₇×#8557, C₃₇×#8559, C₃₇×#8561, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 532);