C₃₇×#1500 #17990318

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×#1500

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	#121×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

238 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 1184

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)(39,42,43,44)(46,47)(48,54,56,51)(49,52)(50,53)(55,60,61,58)(57,59);(38,39)(40,44)(41,43)(42,45)(46,48)(47,51)(49,55)(50,56)(52,58)(53,54)(57,61)(59,60);(46,49,50,57)(47,52,53,59)(48,55,56,61)(51,58,54,60)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8247, C₃₇×#8248, C₃₇×#8249, …
maximal normal subgroup of	C₃₇×#8247, C₃₇×#8248, C₃₇×#8249, …
maximal quotient of	C₃₇×#7560, C₃₇×#7562, C₃₇×#7567, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 232);