C₃₇×#493 #14557158

order	2368
nilpotency class	3
isoclinism	isoclinic class of #465
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₇×#493

Subgroups

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

162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 3 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)(39,41)(43,46)(44,47)(48,49);(42,43,47,49,45,48,44,46);(38,39,40,41)

Conjugacy classes & character table

814 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#1357, C₃₇×#1361, C₃₇×#1408, …
maximal normal subgroup of	C₃₇×#1357, C₃₇×#1361, C₃₇×#1408, …
maximal quotient of	C₃₇×#1706, C₃₇×#1709, C₃₇×#1712, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 174);

C37×#493 #14557158

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₃₇×#493 #14557158