C₃₇×#518 #14557183

order	2368
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#518

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂⁴
commutator	C₂	C₂³×C₄×C₃₇
Frattini, cosocle	C₂²	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ G

338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 4 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)(38,39,40,41)(42,43)(44,48)(45,46)(47,49);(43,46)(47,48);(42,43)(44,47)(45,46)(48,49);(42,44)(43,47)(45,49)(46,48)

Conjugacy classes & character table

1258 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1828, C₃₇×#1829, C₃₇×#1856, …
maximal normal subgroup of	C₃₇×#1828, C₃₇×#1829, C₃₇×#1856, …
maximal quotient of	C₃₇×#2273, C₃₇×#2274, C₃₇×#2275, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 199);

C37×#518 #14557183