C₃₇×#1543 #17990361

order	4736
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1543

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₃₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

246 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 96 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)(38,39,40,43,41,42,44,45)(46,47)(48,55)(49,53)(50,58)(51,54)(52,56)(57,60)(59,61);(39,42)(40,44)(46,48)(47,52)(49,56)(50,59)(51,57)(53,60)(54,61)(55,58);(46,49,51,50)(47,53,54,58)(48,56,57,59)(52,60,61,55)

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7488, C₃₇×#7490, C₃₇×#8241, …
maximal normal subgroup of	C₃₇×#7488, C₃₇×#7490, C₃₇×#8241, …
maximal quotient of	C₃₇×#7563, C₃₇×#7567, C₃₇×#7645, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 275);