C₃₇×#1828 #17990646

order	4736
nilpotency class	3
isoclinism	isoclinic class of #1828
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₃₇ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₃₇×#1828

Subgroups

	subgroup	G/H
center, socle	C₂×C₃₇	#381
commutator	C₂³	C₂²×C₄×C₃₇
Frattini, cosocle	C₂⁴	C₂³×C₃₇
prime radical	C₂²×D₄×C₃₇	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃₇ ◁ C₂⁴×C₃₇ ◁ G

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

normal subgroups only

This group has 296 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)(40,46,48,43)(41,49)(42,45)(44,50)(47,51,53,52);(38,40)(39,43)(41,47)(42,48)(44,51)(45,46)(49,52)(50,53);(38,41)(39,44,45,49)(40,47)(42,50)(43,51,46,52)(48,53)

Conjugacy classes & character table

1073 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#12725, C₃₇×#12726, C₃₇×#12727, …
maximal normal subgroup of	C₃₇×#12725, C₃₇×#12726, C₃₇×#12727, …
maximal quotient of	C₃₇×#8734, C₃₇×#8735, C₃₇×#8736, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 560);