C₃₇×#2243 #17991061

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2218
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₄

Aliases

direct product

C₃₇×#2243

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂³×C₃₇
prime radical	C₂×D₈×C₃₇	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₄²×C₃₇ ◁ C₄×C₈×C₃₇ ◁ G

142 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 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)(38,39)(40,43)(41,42)(44,45)(46,47)(48,57)(49,54)(50,53)(51,59)(52,56)(55,58)(60,61);(39,42)(43,45)(46,47,50,54,51,55,57,61,52,56,60,48,58,59,49,53);(38,40,41,44)(39,43,42,45)(46,48,49,56,58,61,60,55,52,54,57,47,51,53,50,59)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#12376, C₃₇×#12377, C₃₇×#12378, …
maximal normal subgroup of	C₃₇×#12376, C₃₇×#12377, C₃₇×#12378, …
maximal quotient of	C₃₇×#12890, C₃₇×#13149, C₃₇×#13150, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 975);