Group information
Description: | $C_4^2.D_4$ |
Order: | \(128\)\(\medspace = 2^{7} \) |
Exponent: | \(8\)\(\medspace = 2^{3} \) |
Automorphism group: | Group of order \(1024\)\(\medspace = 2^{10} \) (generators) |
Outer automorphisms: | $C_2^4:D_4$, of order \(128\)\(\medspace = 2^{7} \) |
Composition factors: | $C_2$ x 7 |
Nilpotency class: | $3$ |
Derived length: | $2$ |
This group is nonabelian, a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary), and metabelian.
Group statistics
Order | 1 | 2 | 4 | 8 | |
---|---|---|---|---|---|
Elements | 1 | 7 | 56 | 64 | 128 |
Conjugacy classes | 1 | 7 | 24 | 24 | 56 |
Divisions | 1 | 7 | 14 | 6 | 28 |
Autjugacy classes | 1 | 5 | 10 | 2 | 18 |
Dimension | 1 | 2 | 4 | |
---|---|---|---|---|
Irr. complex chars. | 32 | 24 | 0 | 56 |
Irr. rational chars. | 4 | 10 | 14 | 28 |
Minimal Presentations
Permutation degree: | $20$ |
Transitive degree: | $128$ |
Rank: | $2$ |
Inequivalent generating pairs: | $6$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | none | none | none |
Arbitrary | 4 | 6 | 10 |
Constructions
Presentation: | $\langle a, b, c \mid a^{8}=b^{4}=c^{4}=[a,c]=1, b^{a}=bc, c^{b}=c^{3} \rangle$ | |||||||||
Permutation group: | Degree $20$ $\langle(1,2,5,6)(3,4,7,8)(13,14,16,19,17,15,18,20), (1,3,4,6,5,7,8,2)(9,10,11,12) \!\cdots\! \rangle$ | |||||||||
Matrix group: | $\left\langle \left(\begin{array}{rr} 3 & 6 \\ 0 & 3 \end{array}\right), \left(\begin{array}{rr} 7 & 0 \\ 0 & 7 \end{array}\right), \left(\begin{array}{rr} 9 & 0 \\ 0 & 9 \end{array}\right), \left(\begin{array}{rr} 1 & 4 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 8 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 5 & 3 \\ 2 & 3 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/16\Z)$ | |||||||||
$\left\langle \left(\begin{array}{rr} 1 & 12 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 13 & 0 \\ 0 & 13 \end{array}\right), \left(\begin{array}{rr} 5 & 9 \\ 12 & 11 \end{array}\right), \left(\begin{array}{rr} 5 & 0 \\ 0 & 5 \end{array}\right), \left(\begin{array}{rr} 13 & 8 \\ 8 & 5 \end{array}\right), \left(\begin{array}{rr} 1 & 6 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 3 & 2 \\ 8 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/24\Z)$ | ||||||||||
Direct product: | not isomorphic to a non-trivial direct product | |||||||||
Semidirect product: | $(C_4:C_8)$ $\,\rtimes\,$ $C_4$ (2) | $(C_4\times C_8)$ $\,\rtimes\,$ $C_4$ (2) | $(C_4:C_4)$ $\,\rtimes\,$ $C_8$ (4) | more information | ||||||
Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||||
Non-split product: | $C_4^2$ . $Q_8$ | $C_4^2$ . $D_4$ | $C_4$ . $(Q_8:C_4)$ | $C_2$ . $(Q_8:C_8)$ (2) | all 36 |
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
Abelianization: | $C_{4} \times C_{8} $ |
Schur multiplier: | $C_{2}^{3}$ |
Commutator length: | $1$ |
Subgroups
There are 176 subgroups in 104 conjugacy classes, 60 normal (36 characteristic).
Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_2^2\times C_4$ | $G/Z \simeq$ $D_4$ |
Commutator: | $G' \simeq$ $C_4$ | $G/G' \simeq$ $C_4\times C_8$ |
Frattini: | $\Phi \simeq$ $C_2\times C_4^2$ | $G/\Phi \simeq$ $C_2^2$ |
Fitting: | $\operatorname{Fit} \simeq$ $C_4^2.D_4$ | $G/\operatorname{Fit} \simeq$ $C_1$ |
Radical: | $R \simeq$ $C_4^2.D_4$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_2^3$ | $G/\operatorname{soc} \simeq$ $C_2^2:C_4$ |
2-Sylow subgroup: | $P_{ 2 } \simeq$ $C_4^2.D_4$ |
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 | $C_4^2.D_4$ | $\rhd$ | $C_4$ | $\rhd$ | $C_1$ | ||||||||||
Chief series | $C_4^2.D_4$ | $\rhd$ | $C_4:C_4^2$ | $\rhd$ | $C_2\times C_4^2$ | $\rhd$ | $C_2^2\times C_4$ | $\rhd$ | $C_2\times C_4$ | $\rhd$ | $C_2^2$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ |
Lower central series | $C_4^2.D_4$ | $\rhd$ | $C_4$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ | ||||||||
Upper central series | $C_1$ | $\lhd$ | $C_2^2\times C_4$ | $\lhd$ | $C_2\times C_4^2$ | $\lhd$ | $C_4^2.D_4$ |
Supergroups
This group is a maximal subgroup of 284 larger groups in the database.
This group is a maximal quotient of 28 larger groups in the database.
Character theory
Complex character table
See the $56 \times 56$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
See the $28 \times 28$ rational character table.