C₂×C₃₇×#444 #17991772

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×C₃₇×#444

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator	C₄	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂×#135×C₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1184

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

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12650, C₃₇×#12651, C₃₇×#12944, …
maximal normal subgroup of	C₃₇×#12650, C₃₇×#12651, C₃₇×#12944, …
maximal quotient of	C₂×C₃₇×SD₁₆⋊C₈, C₂×C₃₇×#1579, C₂×C₃₇×#1580, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1686);

C2×C37×#444 #17991772