C₃₇×#1773 #17990591

order	4736
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄²×C₃₇ · Schur multiplier (H₂) = C₂×C₄

Aliases

direct product

C₃₇×#1773

Subgroups

	subgroup	G/H
center	C₂×C₄×C₃₇	C₂×D₄
commutator	C₄	C₂×C₄²×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂²×C₃₇	C₄×D₄
prime radical	C₂³×C₃₇	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ C₂²×C₈×C₃₇ ◁ G

224 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 1184

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)(43,47,49,52)(44,50)(48,55,56,54)(51,57);(38,39,40,41)(42,43)(44,48)(45,52)(46,49)(47,53)(50,56)(51,55)(54,57);(42,44,45,51,46,50,53,57)(43,48,52,55,49,56,47,54)

Conjugacy classes & character table

1628 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12647, C₃₇×#12648, C₃₇×#12649, …
maximal normal subgroup of	C₃₇×#12647, C₃₇×#12648, C₃₇×#12649, …
maximal quotient of	C₃₇×#8268, C₃₇×#8269, C₃₇×#8270, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 505);