C₃₇×#1484 #17990302

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

Aliases

direct product

C₃₇×#1484

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
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 ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

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

normal subgroups only

This group has 262 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) = 53
transitive degree	μ_t(G) = 2368

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)(40,44)(41,43)(42,45)(46,47,50,53,51,48,49,52);(39,43)(40,41)(44,45)(46,48)(47,51)(49,53)(50,52);(38,40,42,41)(39,44,45,43)(46,49,51,50)(47,52,48,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8412, C₃₇×#8414, C₃₇×#8483, …
maximal normal subgroup of	C₃₇×#8412, C₃₇×#8414, C₃₇×#8483, …
maximal quotient of	C₃₇×#7674, C₃₇×#7679, C₃₇×#7683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 216);