Group information
Description: | $C_{16}:C_4$ |
Order: | \(64\)\(\medspace = 2^{6} \) |
Exponent: | \(16\)\(\medspace = 2^{4} \) |
Automorphism group: | $D_8.(C_4\times D_4)$, of order \(512\)\(\medspace = 2^{9} \) (generators) |
Outer automorphisms: | $C_4\times D_4$, of order \(32\)\(\medspace = 2^{5} \) |
Composition factors: | $C_2$ x 6 |
Nilpotency class: | $4$ |
Derived length: | $2$ |
This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metacyclic (hence metabelian).
Group statistics
Order | 1 | 2 | 4 | 8 | 16 | |
---|---|---|---|---|---|---|
Elements | 1 | 3 | 36 | 8 | 16 | 64 |
Conjugacy classes | 1 | 3 | 6 | 4 | 8 | 22 |
Divisions | 1 | 3 | 4 | 2 | 2 | 12 |
Autjugacy classes | 1 | 2 | 3 | 2 | 1 | 9 |
Dimension | 1 | 2 | 4 | 8 | |
---|---|---|---|---|---|
Irr. complex chars. | 8 | 14 | 0 | 0 | 22 |
Irr. rational chars. | 4 | 4 | 2 | 2 | 12 |
Minimal Presentations
Permutation degree: | $20$ |
Transitive degree: | $64$ |
Rank: | $2$ |
Inequivalent generating pairs: | $3$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | none | none | none |
Arbitrary | 3 | 6 | 10 |
Constructions
Presentation: | $\langle a, b \mid a^{4}=b^{16}=1, b^{a}=b^{7} \rangle$ | |||||||||
Permutation group: | Degree $20$ $\langle(1,2,5,8)(3,4,11,13)(6,15,12,9)(7,10,16,14)(17,18,19,20), (1,3,10,9,6,7,4,8,5,11,14,15,12,16,13,2) \!\cdots\! \rangle$ | |||||||||
Matrix group: | $\left\langle \left(\begin{array}{rr} 3 & 3 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 1 & 2 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 15 & 11 \\ 8 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$ | |||||||||
$\left\langle \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 4 & 3 \\ 9 & 7 \end{array}\right), \left(\begin{array}{rr} 1 & 12 \\ 15 & 13 \end{array}\right), \left(\begin{array}{rr} 0 & 4 \\ 2 & 0 \end{array}\right), \left(\begin{array}{rr} 8 & 0 \\ 0 & 8 \end{array}\right), \left(\begin{array}{rr} 20 & 6 \\ 18 & 5 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/21\Z)$ | ||||||||||
Direct product: | not isomorphic to a non-trivial direct product | |||||||||
Semidirect product: | $C_{16}$ $\,\rtimes\,$ $C_4$ (2) | more information | ||||||||
Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||||
Non-split product: | $C_8$ . $Q_8$ | $C_4$ . $Q_{16}$ | $C_2^2$ . $D_8$ | $C_2$ . $\SD_{32}$ (2) | all 11 |
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
Abelianization: | $C_{2} \times C_{4} $ |
Schur multiplier: | $C_{2}$ |
Commutator length: | $1$ |
Subgroups
There are 45 subgroups in 23 conjugacy classes, 17 normal (11 characteristic).
Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_2^2$ | $G/Z \simeq$ $D_8$ |
Commutator: | $G' \simeq$ $C_8$ | $G/G' \simeq$ $C_2\times C_4$ |
Frattini: | $\Phi \simeq$ $C_2\times C_8$ | $G/\Phi \simeq$ $C_2^2$ |
Fitting: | $\operatorname{Fit} \simeq$ $C_{16}:C_4$ | $G/\operatorname{Fit} \simeq$ $C_1$ |
Radical: | $R \simeq$ $C_{16}:C_4$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_2^2$ | $G/\operatorname{soc} \simeq$ $D_8$ |
2-Sylow subgroup: | $P_{ 2 } \simeq$ $C_{16}:C_4$ |
Subgroup diagram and profile
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.
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Series
Derived series | $C_{16}:C_4$ | $\rhd$ | $C_8$ | $\rhd$ | $C_1$ | ||||||||
Chief series | $C_{16}:C_4$ | $\rhd$ | $C_2\times C_{16}$ | $\rhd$ | $C_2\times C_8$ | $\rhd$ | $C_8$ | $\rhd$ | $C_4$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ |
Lower central series | $C_{16}:C_4$ | $\rhd$ | $C_8$ | $\rhd$ | $C_4$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ | ||||
Upper central series | $C_1$ | $\lhd$ | $C_2^2$ | $\lhd$ | $C_2\times C_4$ | $\lhd$ | $C_2\times C_8$ | $\lhd$ | $C_{16}:C_4$ |
Supergroups
This group is a maximal subgroup of 61 larger groups in the database.
This group is a maximal quotient of 43 larger groups in the database.
Character theory
Complex character table
See the $22 \times 22$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
1A | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 16A | 16B | ||
Size | 1 | 1 | 1 | 1 | 2 | 2 | 16 | 16 | 4 | 4 | 8 | 8 | |
2 P | 1A | 1A | 1A | 1A | 2A | 2A | 2C | 2B | 4B | 4B | 8B | 8B | |
Schur | |||||||||||||
64.48.1a | |||||||||||||
64.48.1b | |||||||||||||
64.48.1c | |||||||||||||
64.48.1d | |||||||||||||
64.48.1e | |||||||||||||
64.48.1f | |||||||||||||
64.48.2a | |||||||||||||
64.48.2b | 2 | ||||||||||||
64.48.2c | |||||||||||||
64.48.2d | 2 | ||||||||||||
64.48.2e | |||||||||||||
64.48.2f |