C₃₇×#2975 #17991793

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

Aliases

direct product

C₃₇×#2975

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	C₂²×D₄×C₃₇	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₇ ◁ G

452 conjugacy classes of subgroups; only the 268 normal subgroups are shown.

normal subgroups only

This group has 452 conjugacy classes of subgroups — too many to draw. Only its 268 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) = 1184

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

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃₇×#8837, C₃₇×#8860, C₃₇×#8961, …
maximal normal subgroup of	C₃₇×#8837, C₃₇×#8860, C₃₇×#8961, …
maximal quotient of	C₃₇×#20524, C₃₇×#20536, C₃₇×#20568, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 1707);