C₂×C₃₇×#398 #17991209

order	4736
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₃₇×#398

Subgroups

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

692 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 4736

A minimal faithful permutation representation — 4 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);(7,8,9,10)(12,15)(16,17);(4,6)(11,12,14,15)(13,17,18,16);(3,4)(5,6)(7,8)(9,10)(11,13,14,18)(12,16,15,17)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9026, C₃₇×#9027, C₃₇×#9919, …
maximal normal subgroup of	C₃₇×#9026, C₃₇×#9027, C₃₇×#9919, …
maximal quotient of	C₂×C₃₇×#1441, C₂×C₃₇×#1442, C₂×C₃₇×#1444, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1123);

C2×C37×#398 #17991209