C₂×C₃₇×#478 #17991890

order	4736
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×C₃₇×#478

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 2368

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

Conjugacy classes & character table

1406 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9245, C₄×C₃₇×#478, C₃₇×#21617, …
maximal normal subgroup of	C₃₇×#9245, C₄×C₃₇×#478, C₃₇×#21617, …
maximal quotient of	C₂×C₃₇×#1804, C₂×C₃₇×#1862, C₂×C₃₇×#1864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1804);

C2×C37×#478 #17991890