C₃₇×#1502 #17990320

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

Aliases

direct product

C₃₇×#1502

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	C₃₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

284 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 53
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 generators on 53 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)(41,44)(43,45)(47,51,52,53);(39,42)(40,45)(43,44)(46,47)(48,51)(49,53)(50,52);(46,48,50,49)(47,51,52,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8251, C₃₇×#8253, C₃₇×#8255, …
maximal normal subgroup of	C₃₇×#8251, C₃₇×#8253, C₃₇×#8255, …
maximal quotient of	C₃₇×#7560, C₃₇×#7569, C₃₇×#7570, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 234);