C₃₇×#3259 #17992077

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

Aliases

direct product

C₃₇×#3259

Subgroups

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

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

normal subgroups only

This group has 370 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) = 61
transitive degree	μ_t(G) = 2368

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

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12164, C₃₇×#12167, C₃₇×#12170, …
maximal normal subgroup of	C₃₇×#12164, C₃₇×#12167, C₃₇×#12170, …
maximal quotient of	C₃₇×#21897, C₃₇×#21968, C₃₇×#22126, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1991);