C₂×C₃₇×#330 #17990341

order	4736
nilpotency class	3
isoclinism	isoclinic class of #327
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₂×C₃₇×#330

Subgroups

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

232 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 4736

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄×C₃₇×#330, C₃₇×#8225, C₃₇×#8227, …
maximal normal subgroup of	C₄×C₃₇×#330, C₃₇×#8225, C₃₇×#8227, …
maximal quotient of	C₂×C₃₇×#1271, C₃₇×#7568, C₃₇×#7627, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 255);

C2×C37×#330 #17990341