C₃₇×#2971 #17991789

order	4736
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₃₇×#2971

Subgroups

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

402 conjugacy classes of subgroups; only the 276 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 276 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) = 2368

A minimal faithful permutation representation — 4 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)(40,43)(41,42)(44,45)(46,47,51,48,50,52,49,53);(39,42)(43,45)(47,52)(48,53);(46,48,51,52,50,53,49,47);(38,40,41,44)(39,43,42,45)(46,49,50,51)(47,48,52,53)

Conjugacy classes & character table

1850 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#9160, C₃₇×#9162, C₃₇×#9350, …
maximal normal subgroup of	C₃₇×#9160, C₃₇×#9162, C₃₇×#9350, …
maximal quotient of	C₃₇×#20480, C₃₇×#20494, Q₈×#97×C₃₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1703);