C₂×C₃₇×#386 #17991105

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×C₃₇×#386

Subgroups

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

1576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 51 points:

(1,2)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51);(4,6)(7,8,9,10)(12,14);(7,8)(9,10)(11,12,13,14);(3,4)(5,6)(7,8)(9,10)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9079, C₃₇×#9083, C₃₇×#10745, …
maximal normal subgroup of	C₃₇×#9079, C₃₇×#9083, C₃₇×#10745, …
maximal quotient of	C₂×C₃₇×C₂²⋊#95, C₂×C₃₇×#1436, C₂×C₃₇×#1437, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1019);

C2×C37×#386 #17991105