C₃₇×#2228 #17991046

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

Aliases

direct product

C₃₇×#2228

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

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

normal subgroups only

This group has 136 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)(38,39,41,43)(40,44,45,42)(46,47,48,51)(49,52,55,59)(50,58,56,54)(53,61,60,57);(38,40,41,45)(39,42,43,44)(47,51)(52,59)(53,60)(54,58);(47,52)(49,57)(50,56)(51,59)(53,54)(55,61)(58,60)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃₇×#2228, C₃₇×#33626, C₃₇×#33627, …
maximal normal subgroup of	C₂×C₃₇×#2228, C₃₇×#33626, C₃₇×#33627, …
maximal quotient of	C₃₇×#12813, C₃₇×#12935, C₃₇×#12937, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 960);