C₃₇×#1735 #17990553

order	4736
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₃₇×#1735

Subgroups

	subgroup	G/H
center	C₂²×C₄×C₃₇	D₄
commutator	C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂³×C₃₇	C₂×D₄
prime radical	C₂⁴×C₃₇	D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

424 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 232 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 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)(40,41)(42,43,44,45)(46,47)(48,56)(49,53)(50,54)(51,55)(52,57)(58,61)(59,60);(39,41)(46,48,50,58,51,59,49,57)(47,52,53,60,55,61,54,56);(46,49,51,50)(47,53,55,54)(48,57,59,58)(52,60,61,56)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7260, C₃₇×#7262, C₃₇×#10706, …
maximal normal subgroup of	C₃₇×#7260, C₃₇×#7262, C₃₇×#10706, …
maximal quotient of	C₃₇×#7872, C₃₇×#7875, C₃₇×#7878, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 467);