Group information
| Description: | $C_4^2.D_4$ |
| Order: | \(128\)\(\medspace = 2^{7} \) |
| Exponent: | \(8\)\(\medspace = 2^{3} \) |
| Automorphism group: | $C_2^4.C_2^2\wr C_3$, of order \(3072\)\(\medspace = 2^{10} \cdot 3 \) (generators) |
| Outer automorphisms: | $\SL(2,3):C_2^2$, of order \(96\)\(\medspace = 2^{5} \cdot 3 \) |
| 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 | 24 | 96 | 128 |
| Conjugacy classes | 1 | 4 | 9 | 12 | 26 |
| Divisions | 1 | 4 | 6 | 6 | 17 |
| Autjugacy classes | 1 | 2 | 2 | 1 | 6 |
| Dimension | 1 | 2 | 4 | 8 | |
|---|---|---|---|---|---|
| Irr. complex chars. | 16 | 4 | 6 | 0 | 26 |
| Irr. rational chars. | 4 | 10 | 0 | 3 | 17 |
Minimal Presentations
| Permutation degree: | $32$ |
| Transitive degree: | $64$ |
| Rank: | $2$ |
| Inequivalent generating pairs: | $2$ |
Minimal degrees of faithful linear representations
| Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
|---|---|---|---|
| Irreducible | none | none | none |
| Arbitrary | 8 | 16 | 16 |
Constructions
| Presentation: |
$\langle a, b, c, d \mid b^{2}=c^{2}=d^{8}=[a,c]=[b,c]=[c,d]=1, a^{4}=cd^{4}, b^{a}=bd^{4}, d^{a}=bcd, d^{b}=cd \rangle$
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| Permutation group: | Degree $32$
$\langle(1,2)(3,6,9,13)(4,11)(5,7)(8,12)(10,16,15,14)(17,18,21,26,23,19,28,30)(20,29,22,27,32,25,24,31) \!\cdots\! \rangle$
| |||||||
| Direct product: | not isomorphic to a non-trivial direct product | |||||||
| Semidirect product: | not isomorphic to a non-trivial semidirect product | |||||||
| Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||
| Non-split product: | $C_4^2$ . $Q_8$ | $C_4^2$ . $D_4$ (3) | $C_2^3$ . $C_4^2$ | $(C_4^2.C_4)$ . $C_2$ (3) | all 11 | |||
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
| Abelianization: | $C_{4}^{2} $ |
| Schur multiplier: | $C_{2}$ |
| Commutator length: | $1$ |
Subgroups
There are 112 subgroups in 61 conjugacy classes, 30 normal (6 characteristic).
Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.
Special subgroups
| Center: | $Z \simeq$ $C_2^2$ | $G/Z \simeq$ $C_2.C_4^2$ |
| Commutator: | $G' \simeq$ $C_2^3$ | $G/G' \simeq$ $C_4^2$ |
| 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: | $\operatorname{soc} \simeq$ $C_2^2$ | $G/\operatorname{soc} \simeq$ $C_2.C_4^2$ |
| 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 | $C_4^2.D_4$ | $\rhd$ | $C_2^3$ | $\rhd$ | $C_1$ | ||||||||||
| Chief series | $C_4^2.D_4$ | $\rhd$ | $C_4^2.C_4$ | $\rhd$ | $C_2\times C_4^2$ | $\rhd$ | $C_2^2\times C_4$ | $\rhd$ | $C_2\times C_4$ | $\rhd$ | $C_2^2$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ |
| Lower central series | $C_4^2.D_4$ | $\rhd$ | $C_2^3$ | $\rhd$ | $C_2^2$ | $\rhd$ | $C_1$ | ||||||||
| Upper central series | $C_1$ | $\lhd$ | $C_2^2$ | $\lhd$ | $C_2\times C_4^2$ | $\lhd$ | $C_4^2.D_4$ |
Supergroups
This group is a maximal subgroup of 17 larger groups in the database.
This group is a maximal quotient of 6 larger groups in the database.
Character theory
Complex character table
See the $26 \times 26$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
See the $17 \times 17$ rational character table.