C₃₇×#3416 #17992234

order	4736
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₃₇×#3416

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂²×D₈
commutator, Frattini, cosocle	C₈	C₂⁴×C₃₇
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ #41×C₃₇ ◁ #131×C₃₇ ◁ G

360 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 69
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 69 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)(38,39,43,48)(40,45,49,59)(41,44,54,58)(42,55,53,47)(46,62,57,52)(50,56,63,67)(51,66,64,61)(60,69,68,65);(39,44)(41,52)(42,53)(45,56)(46,47)(48,58)(50,65)(51,64)(54,62)(55,57)(59,67)(60,61)(63,69)(66,68);(38,40)(39,45)(41,50)(42,51)(43,49)(44,56)(46,60)(47,61)(48,59)(52,65)(53,64)(54,63)(55,66)(57,68)(58,67)(62,69);(40,49)(45,59)(50,63)(51,64)(56,67)(60,68)(61,66)(65,69)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#33854, C₃₇×#33855, C₃₇×#33856, …
maximal normal subgroup of	C₃₇×#33854, C₃₇×#33855, C₃₇×#33856, …
maximal quotient of	C₃₇×#33381, C₃₇×#33382, C₃₇×#33411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2148);