C₂×C₃₇×#556 #17992292

order	4736
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×C₃₇×#556

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

1112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8365, C₃₇×#8368, C₃₇×#8389, …
maximal normal subgroup of	C₃₇×#8365, C₃₇×#8368, C₃₇×#8389, …
maximal quotient of	C₂×C₃₇×#2347, C₂×C₃₇×#2354, C₂×C₃₇×#2355, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2206);

C2×C37×#556 #17992292