C₃₇×#2226 #17991044

order	4736
nilpotency class	4
isoclinism	isoclinic class of #2184
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2226

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 68 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 68 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)(38,39,41,43)(40,44,45,42)(46,47,50,54)(48,51,57,60)(49,58,56,53)(52,61,59,55);(38,40,41,45)(39,42,43,44);(47,51)(48,55)(49,56)(52,53)(54,60)(57,61)(58,59)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12372, C₃₇×#12375, C₃₇×#12378, …
maximal normal subgroup of	C₃₇×#12372, C₃₇×#12375, C₃₇×#12378, …
maximal quotient of	C₃₇×#12811, C₃₇×#12937, C₃₇×#13111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 958);