C₃₇×#1827 #17990645

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1827

Subgroups

	subgroup	G/H
center	C₄×C₃₇	#117
commutator	C₂×C₄	C₂²×C₄×C₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₃₇
socle	C₂×C₃₇	#381
prime radical	C₂×#41×C₃₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₇ ◁ #117×C₃₇ ◁ G

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

normal subgroups only

This group has 262 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) = 53
transitive degree	μ_t(G) = 592

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#12719, C₃₇×#12720, C₃₇×#12721, …
maximal normal subgroup of	C₃₇×#12719, C₃₇×#12720, C₃₇×#12721, …
maximal quotient of	C₃₇×#8730, C₃₇×#8731, C₃₇×#8732, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 559);