C₃₇×#3022 #17991840

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₇×#3022

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂×#120
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₃₇×#585	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ G

704 conjugacy classes of subgroups; only the 212 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 212 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)(38,39,44,49)(40,50,53,62)(41,46,55,61)(42,48,43,47)(45,56,59,52)(51,65,68,64)(54,63,57,60)(58,66,69,67);(38,40)(39,45)(41,51)(42,52)(43,56)(44,53)(46,58)(47,50)(48,62)(49,59)(54,66)(55,68)(57,67)(60,64)(61,69)(63,65);(38,41)(39,46)(40,51)(42,54)(43,57)(44,55)(45,58)(47,60)(48,63)(49,61)(50,64)(52,66)(53,68)(56,67)(59,69)(62,65);(38,42,44,43)(39,47,49,48)(40,52,53,56)(41,54,55,57)(45,50,59,62)(46,60,61,63)(51,66,68,67)(58,64,69,65)

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#13355, C₃₇×#13356, C₃₇×#13361, …
maximal normal subgroup of	C₃₇×#13355, C₃₇×#13356, C₃₇×#13361, …
maximal quotient of	C₃₇×#18136, C₃₇×#18140, C₃₇×#18146, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1754);