C₂²×#120×C₃₇ #17992249

order	4736
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃₇ · Schur multiplier (H₂) = C₂¹¹

Aliases

direct product

C₂²×#120×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂⁴×C₃₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₇ ◁ G

4544 conjugacy classes of subgroups; only the 1112 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 1112 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) = 49
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 5 generators on 49 points:

(1,2)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49);(3,4);(5,6)(7,9)(8,10)(11,12);(6,9)(10,12);(6,10)(9,12)

Conjugacy classes & character table

2072 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#8541, C₃₇×#8547, C₃₇×#10965, …
maximal normal subgroup of	C₃₇×#8541, C₃₇×#8547, C₃₇×#10965, …
maximal quotient of	C₂²×C₃₇×#379, C₂²×C₃₇×#385, C₂²×C₃₇×#386, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4736, 2163);