C₃₇×#2237 #17991055

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₃₇×#2237

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
prime radical	C₂³×C₃₇	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

132 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 77
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 77 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)(40,43)(41,42)(44,45)(46,47,51,56)(48,52,59,67)(49,53,62,69)(50,63,61,55)(54,70,68,60)(57,64,73,76)(58,74,72,66)(65,77,75,71);(39,42)(43,45)(46,48,51,59)(47,52,56,67)(49,57,62,73)(50,58,61,72)(53,64,69,76)(54,65,68,75)(55,66,63,74)(60,71,70,77);(38,40,41,44)(39,43,42,45)(47,53)(49,60)(50,61)(52,64)(54,55)(56,69)(57,71)(58,72)(62,70)(63,68)(65,66)(67,76)(73,77)(74,75)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×#2237, C₃₇×#33645, C₃₇×#33647, …
maximal normal subgroup of	C₂×C₃₇×#2237, C₃₇×#33645, C₃₇×#33647, …
maximal quotient of	C₃₇×#12868, C₃₇×#12921, C₃₇×#12994, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 969);