C₃₇×#2159 #17990977

order	4736
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂²

Aliases

direct product

C₃₇×#2159

Subgroups

	subgroup	G/H
center, socle	C₂²×C₃₇	C₂×D₈
commutator	C₈	C₂²×C₄×C₃₇
Frattini, cosocle	C₂×C₈	C₂³×C₃₇
prime radical	C₂³×C₃₇	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂²×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

156 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 156 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 57
transitive degree	μ_t(G) = 2368

A minimal faithful permutation representation — 3 generators on 57 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)(38,39,40,41)(43,47)(44,48)(45,51)(49,53)(50,57)(52,56)(54,55);(42,43,48,55,45,50,52,53,46,49,56,57,51,54,44,47);(43,49)(47,53)(50,54)(55,57)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12677, C₃₇×#12923, C₃₇×#12925, …
maximal normal subgroup of	C₃₇×#12677, C₃₇×#12923, C₃₇×#12925, …
maximal quotient of	C₃₇×#12009, C₃₇×#12025, C₃₇×#12035, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 891);