C₂×C₃₇×#334 #17990380

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×C₃₇×#334

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	D₄
commutator	C₄	C₂²×C₈×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
socle, prime radical	C₂³×C₃₇	C₄⋊C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

200 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 55
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 3 generators on 55 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(4,7)(5,9)(8,10)(11,12,13,15,14,16,17,18);(3,4,5,7,6,8,9,10)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7156, C₃₇×#7158, C₃₇×#7173, …
maximal normal subgroup of	C₃₇×#7156, C₃₇×#7158, C₃₇×#7173, …
maximal quotient of	C₂×C₃₇×#1277, C₄×C₃₇×#334, C₃₇×#8595, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 294);