C₂×C₃₇×#322 #17990266

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈×C₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×C₃₇×#322

Subgroups

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

248 conjugacy classes of subgroups.

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);(3,4,5,8,6,7,9,10);(4,7)(8,10)(11,12,13,15,14,16,17,18)

Conjugacy classes & character table

2960 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#7137, C₃₇×#7139, C₃₇×#7145, …
maximal normal subgroup of	C₃₇×#7137, C₃₇×#7139, C₃₇×#7145, …
maximal quotient of	C₄×C₃₇×#322, C₂×C₃₇×#1273, C₃₇×#8131, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 180);

C2×C37×#322 #17990266