C₃₇×#3342 #17992160

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3342

Subgroups

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

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 61
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 4 generators on 61 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37)(38,39,42,41)(40,44,45,43)(46,47)(48,52)(49,56)(50,58)(51,54)(53,55)(57,60)(59,61);(38,40)(39,43)(41,44)(42,45)(46,48,51,50)(47,52,54,58)(49,55,57,59)(53,60,61,56);(38,41,42,39)(40,44,45,43)(46,49,51,57)(47,53,54,61)(48,55,50,59)(52,60,58,56);(40,45)(43,44)(46,48,51,50)(47,52,54,58)(49,55,57,59)(53,60,61,56)

Conjugacy classes & character table

1295 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12341, C₃₇×#12345, C₃₇×#12393, …
maximal normal subgroup of	C₃₇×#12341, C₃₇×#12345, C₃₇×#12393, …
maximal quotient of	C₃₇×#22490, C₃₇×#22614, C₃₇×#22658, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2074);

C37×#3342 #17992160