C₂²×#137×C₃₇ #17992397

order	4736
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#137×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂×D₄
commutator, Frattini, cosocle	C₄	C₂⁵×C₃₇
prime radical	C₂²×#41×C₃₇	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

1464 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 5 generators on 57 points:

(1,2)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(3,4);(5,6)(7,13)(8,16)(9,11)(10,14)(12,18)(15,20)(17,19);(5,7,9,15)(6,10,11,19)(8,17,18,14)(12,20,16,13);(6,11)(10,19)(12,16)(13,20)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#11013, C₃₇×#11015, C₃₇×#11017, …
maximal normal subgroup of	C₃₇×#11013, C₃₇×#11015, C₃₇×#11017, …
maximal quotient of	C₂²×C₃₇×#419, C₂²×C₃₇×#417, C₂²×C₃₇×#428, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2311);

C22×#137×C37 #17992397