C₃₇×#1798 #17990616

order	4736
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1798

Subgroups

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

312 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 61
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 61 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)(44,45)(46,47,49,52,50,53,56,59)(48,51,55,58,54,57,60,61);(38,39,40,41)(46,48)(47,51)(49,55)(50,54)(52,58)(53,57)(56,60)(59,61);(43,45)(48,54)(51,57)(55,60)(58,61)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7252, C₃₇×#7253, C₃₇×#8409, …
maximal normal subgroup of	C₃₇×#7252, C₃₇×#7253, C₃₇×#8409, …
maximal quotient of	C₃₇×#8561, C₃₇×#8562, C₃₇×#8567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 530);