Group information
| Description: | $C_5^2$ |
| Order: | \(25\)\(\medspace = 5^{2} \) |
| Exponent: | \(5\) |
| Automorphism group: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) (generators) |
| Outer automorphisms: | $\GL(2,5)$, of order \(480\)\(\medspace = 2^{5} \cdot 3 \cdot 5 \) |
| Composition factors: | $C_5$ x 2 |
| Nilpotency class: | $1$ |
| Derived length: | $1$ |
This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), a $p$-group (hence elementary and hyperelementary), and metacyclic.
Group statistics
| Order | 1 | 5 | |
|---|---|---|---|
| Elements | 1 | 24 | 25 |
| Conjugacy classes | 1 | 24 | 25 |
| Divisions | 1 | 6 | 7 |
| Autjugacy classes | 1 | 1 | 2 |
| Dimension | 1 | 4 | |
|---|---|---|---|
| Irr. complex chars. | 25 | 0 | 25 |
| Irr. rational chars. | 1 | 6 | 7 |
Minimal Presentations
| Permutation degree: | $10$ |
| Transitive degree: | $25$ |
| Rank: | $2$ |
| Inequivalent generating pairs: | $1$ |
Minimal degrees of faithful linear representations
| Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
|---|---|---|---|
| Irreducible | none | none | none |
| Arbitrary | 2 | 4 | 8 |
Constructions
| Presentation: | Abelian group
$\langle a, b \mid a^{5}=b^{5}=1 \rangle$
| |||||||||
| Permutation group: | Degree $10$
$\langle(1,5,4,3,2), (6,10,9,8,7)\rangle$
| |||||||||
| Matrix group: | $\left\langle \left(\begin{array}{rr} 4 & 0 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 4 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{11})$ | |||||||||
| Transitive group: | 25T2 | more information | ||||||||
| Direct product: | $C_5$ ${}^2$ | |||||||||
| Semidirect product: | not isomorphic to a non-trivial semidirect product | |||||||||
| Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||||
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
| Primary decomposition: | $C_{5}^{2}$ |
| Schur multiplier: | $C_{5}$ |
| Commutator length: | $0$ |
Subgroups
There are 8 subgroups, all normal (2 characteristic).
Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.
Special subgroups
| Center: | $Z \simeq$ $C_5^2$ | $G/Z \simeq$ $C_1$ |
| Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_5^2$ |
| Frattini: | $\Phi \simeq$ $C_1$ | $G/\Phi \simeq$ $C_5^2$ |
| Fitting: | $\operatorname{Fit} \simeq$ $C_5^2$ | $G/\operatorname{Fit} \simeq$ $C_1$ |
| Radical: | $R \simeq$ $C_5^2$ | $G/R \simeq$ $C_1$ |
| Socle: | $\operatorname{soc} \simeq$ $C_5^2$ | $G/\operatorname{soc} \simeq$ $C_1$ |
| 5-Sylow subgroup: | $P_{ 5 } \simeq$ $C_5^2$ |
Subgroup diagram and profile
For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.
To see subgroups sorted vertically by order instead, check this box.
Subgroup information
Click on a subgroup in the diagram to see information about it.
|
Series
| Derived series | $C_5^2$ | $\rhd$ | $C_1$ | ||
| Chief series | $C_5^2$ | $\rhd$ | $C_5$ | $\rhd$ | $C_1$ |
| Lower central series | $C_5^2$ | $\rhd$ | $C_1$ | ||
| Upper central series | $C_1$ | $\lhd$ | $C_5^2$ |
Supergroups
This group is a maximal subgroup of 53 larger groups in the database.
This group is a maximal quotient of 51 larger groups in the database.
Character theory
Complex character table
See the $25 \times 25$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
| 1A | 5A | 5B | 5C | 5D | 5E | 5F | ||
| Size | 1 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 5 P | 1A | 5B | 5F | 5F | 5D | 5B | 5E | |
| 25.2.1a | ||||||||
| 25.2.1b | ||||||||
| 25.2.1c | ||||||||
| 25.2.1d | ||||||||
| 25.2.1e | ||||||||
| 25.2.1f | ||||||||
| 25.2.1g |