C₃₇×#3185 #17992003

order	4736
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#3185

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂³×C₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂×C₄²×C₃₇ ◁ G

358 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 184 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)(38,39)(40,43)(41,42)(44,45)(46,47,50,53)(48,54,55,61)(49,58,56,52)(51,59,60,57);(39,42)(43,45)(48,55)(51,60)(54,61)(57,59);(46,48,56,59,50,55,49,57)(47,51,58,61,53,60,52,54);(38,40,41,44)(39,43,42,45)(48,55)(51,60)(54,61)(57,59)

Conjugacy classes & character table

1184 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9416, C₃₇×#11838, C₃₇×#18557, …
maximal normal subgroup of	C₃₇×#9416, C₃₇×#11838, C₃₇×#18557, …
maximal quotient of	C₃₇×#20824, C₃₇×#21287, C₃₇×#21409, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1917);