Groups of order 2160: Difference between revisions
(Created page with "{{groups of order|2160}} ==Statistics at a glance== The number 2160 has the prime factorization: <math>\! 2160 = 2^4 \cdot 3^3 \cdot 5^1 = 16 \cdot 27 \cdot 5</math> All g...") |
(→Statistics at a glance: https://archive.fo/2dRcX) |
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! Quantity !! Value !! Explanation | ! Quantity !! Value !! Explanation | ||
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| Total number of groups up to isomorphism || | | Total number of groups up to isomorphism || 3429 || | ||
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| Number of [[abelian group]]s (i.e., [[finite abelian group]]s) up to isomorphism || [[abelian count::15]] || (Number of abelian groups of order <math>2^4</math>) times (Number of abelian groups of order <math>3^3</math>) times (Number of abelian groups of order <math>5^1</math>) = ([[number of unordered integer partitions]] of 4) times ([[number of unordered integer partitions]] of 3) times ([[number of unordered integer partitions]] of 1) = <math>5 \times 3 \times 1 = 15</math>. {{abelian count explanation}} | | Number of [[abelian group]]s (i.e., [[finite abelian group]]s) up to isomorphism || [[abelian count::15]] || (Number of abelian groups of order <math>2^4</math>) times (Number of abelian groups of order <math>3^3</math>) times (Number of abelian groups of order <math>5^1</math>) = ([[number of unordered integer partitions]] of 4) times ([[number of unordered integer partitions]] of 3) times ([[number of unordered integer partitions]] of 1) = <math>5 \times 3 \times 1 = 15</math>. {{abelian count explanation}} | ||
Revision as of 11:14, 6 March 2022
This article gives information about, and links to more details on, groups of order 2160
See pages on algebraic structures of order 2160 | See pages on groups of a particular order
Statistics at a glance
The number 2160 has the prime factorization:
All groups of this order have not yet been classified. The information below is therefore partial.
| Quantity | Value | Explanation |
|---|---|---|
| Total number of groups up to isomorphism | 3429 | |
| Number of abelian groups (i.e., finite abelian groups) up to isomorphism | 15 | (Number of abelian groups of order ) times (Number of abelian groups of order ) times (Number of abelian groups of order ) = (number of unordered integer partitions of 4) times (number of unordered integer partitions of 3) times (number of unordered integer partitions of 1) = . See classification of finite abelian groups and structure theorem for finitely generated abelian groups. |
| Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism | 70 | (Number of groups of order 16) times (Number of groups of order 27) times (Number of groups of order 5) = . See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group. |
| Number of solvable groups (i.e., finite solvable groups) up to isomorphism | unknown, but likely between 1000 and 10000 | |
| Number of non-solvable groups up to isomorphism | PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE] | There are two possibilities for the composition factors of non-solvable groups: First: alternating group:A5 (order 60) as the only simple non-abelian composition factor, cyclic group:Z2 (2 times), cyclic group:Z3 (2 times) Second: alternating group:A6 (order 360) as the only simple non-abelian composition factor, cyclic group:Z2 (1 time), cyclic group:Z3 (1 time) |
| Number of simple groups up to isomorphism | 0 | |
| Number of almost simple groups up to isomorphism | 0 | |
| Number of quasisimple groups up to isomorphism | 1 | Schur cover of alternating group:A6 |