C₃₇×#1467 #17990285

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1467

Subgroups

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

236 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 100 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)(38,39,41,43,42,44,40,45)(46,47)(48,54,56,51)(49,52)(50,53)(55,60,61,58)(57,59);(46,48,50,56)(47,51,53,54)(49,55,57,61)(52,58,59,60);(38,40,42,41)(39,43,44,45)(46,49,50,57)(47,52,53,59)(48,55,56,61)(51,58,54,60)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#7248, C₃₇×#7323, C₃₇×#8368, …
maximal normal subgroup of	C₃₇×#7248, C₃₇×#7323, C₃₇×#8368, …
maximal quotient of	C₃₇×#7619, C₃₇×#7622, C₃₇×#7625, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 199);