C₃₇×#3261 #17992079

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3261

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ #117×C₃₇ ◁ G

322 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 168 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 — 4 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)(39,42)(40,44)(46,47,51,55)(48,52,57,66)(49,59,61,69)(50,62,64,54)(53,63,68,60)(56,70,71,77)(58,74,73,67)(65,75,76,72);(38,39)(40,43)(41,42)(44,45)(46,48,51,57)(47,52,55,66)(49,56,61,71)(50,58,64,73)(53,65,68,76)(54,67,62,74)(59,70,69,77)(60,72,63,75);(39,42)(43,45)(46,49,51,61)(47,53,55,68)(48,56,57,71)(50,63,64,60)(52,65,66,76)(54,69,62,59)(58,75,73,72)(67,77,74,70);(38,40,41,44)(39,43,42,45)(48,57)(52,66)(56,71)(58,73)(65,76)(67,74)(70,77)(72,75)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12165, C₃₇×#12168, C₃₇×#12171, …
maximal normal subgroup of	C₃₇×#12165, C₃₇×#12168, C₃₇×#12171, …
maximal quotient of	C₃₇×#21898, C₃₇×#21968, C₃₇×#22124, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1993);