C₃₇×#2366 #17991184

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₇×#2366

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂³	C₂⁴×C₃₇
prime radical	C₂×#127×C₃₇	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

832 conjugacy classes of subgroups; only the 328 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 328 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) = 49
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 49 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)(38,39,41,40)(43,45)(46,47,49,48);(39,40)(46,48,49,47);(42,43,44,45);(38,40,41,39)

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8920, C₃₇×#8921, C₃₇×#8924, …
maximal normal subgroup of	C₃₇×#8920, C₃₇×#8921, C₃₇×#8924, …
maximal quotient of	C₄⋊C₄²×C₃₇, C₃₇×#13820, C₃₇×#13824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1098);