C₂²×#41×C₃₇ #14557247

order	2368
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#41×C₃₇

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	C₂²
commutator, Frattini, cosocle	C₂	C₂⁵×C₃₇
socle	C₂³×C₃₇	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ G

890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 49 points:

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

Conjugacy classes & character table

1480 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1780, C₃₇×#1782, C₃₇×#1785, …
maximal normal subgroup of	C₃₇×#1780, C₃₇×#1782, C₃₇×#1785, …
maximal quotient of	C₂²×#117×C₃₇, C₂²×C₄×D₄×C₃₇, C₂²×C₄×Q₈×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 263);

C22×#41×C37 #14557247