C₃₇×#1533 #17990351

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1533

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₄²	C₂³×C₃₇
prime radical	C₂²×D₄×C₃₇	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

248 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 1184

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)(48,54,56,61)(49,53)(50,52)(51,55,59,57)(58,60);(39,42)(40,44)(46,48)(47,51)(49,57)(50,56)(52,59)(53,61)(54,60)(55,58);(47,52)(51,59)(53,60)(54,61)

Conjugacy classes & character table

962 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃₇×#7481, C₃₇×#7495, C₃₇×#8522, …
maximal normal subgroup of	C₃₇×#7481, C₃₇×#7495, C₃₇×#8522, …
maximal quotient of	C₃₇×#7592, C₃₇×#7595, C₃₇×#7601, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 265);