C₃₇×#1564 #17990382

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1564

Subgroups

	subgroup	G/H
center	C₄²×C₃₇	D₄
commutator	C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

180 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 128 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)(38,39,40,42,41,43,44,45)(46,47)(48,55)(49,53)(50,59)(51,54)(52,57)(56,60)(58,61);(46,48,50,57,51,58,49,56)(47,52,53,55,54,60,59,61);(46,49,51,50)(47,53,54,59)(48,56,58,57)(52,55,60,61)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7155, C₃₇×#7157, C₃₇×#7158, …
maximal normal subgroup of	C₃₇×#7155, C₃₇×#7157, C₃₇×#7158, …
maximal quotient of	C₃₇×#7872, C₄×C₃₇×#334, C₄×C₃₇×#335, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 296);