C₃₇×#2889 #17991707

order	4736
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₃₇×#2889

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₄×C₃₇
Frattini, cosocle	C₂×C₄	C₂⁴×C₃₇
prime radical	C₂×2₋¹⁺⁴×C₃₇	C₂
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂×#41×C₃₇ ◁ G

704 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 340 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) = 69
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 69 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)(38,39,42,47,43,46,55,60)(40,48,50,61,51,44,54,57)(41,45,53,59,52,58,65,68)(49,62,64,69,63,56,66,67);(38,40,43,51)(39,44,46,48)(41,49,52,63)(42,54,55,50)(45,56,58,62)(47,61,60,57)(53,66,65,64)(59,69,68,67);(38,41)(39,45)(40,49)(42,53)(43,52)(44,56)(46,58)(47,59)(48,62)(50,64)(51,63)(54,66)(55,65)(57,67)(60,68)(61,69);(41,52)(45,58)(49,63)(53,65)(56,62)(59,68)(64,66)(67,69)

Conjugacy classes & character table

1517 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#13096, C₃₇×#13098, C₃₇×#21780, …
maximal normal subgroup of	C₃₇×#13096, C₃₇×#13098, C₃₇×#21780, …
maximal quotient of	C₃₇×#17351, C₃₇×#17352, C₃₇×#17368, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1621);