C₃₇×#2122 #17990940

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₃₇×#2122

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂×#99
commutator	C₂³	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×D₄	C₂³×C₃₇
prime radical	C₃₇×#535	C₂
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×C₄×C₃₇ ◁ C₂×#41×C₃₇ ◁ G

266 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 53
transitive degree	μ_t(G) = 592

A minimal faithful permutation representation — 3 generators on 53 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)(40,44)(41,48,49,43)(42,45)(46,52)(47,51,53,50);(39,43)(40,46)(44,51)(45,48)(47,53)(50,52);(38,40,42,46)(39,44,45,52)(41,47,49,53)(43,50,48,51)

Conjugacy classes & character table

851 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12715, C₃₇×#12718, C₃₇×#12721, …
maximal normal subgroup of	C₃₇×#12715, C₃₇×#12718, C₃₇×#12721, …
maximal quotient of	C₃₇×#11427, C₃₇×#11454, C₃₇×#11473, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 854);