Group information
Description: | $(C_2^2\times C_8).D_4$ |
Order: | \(256\)\(\medspace = 2^{8} \) |
Exponent: | \(16\)\(\medspace = 2^{4} \) |
Automorphism group: | $C_4\times D_4^2:C_2^2$, of order \(1024\)\(\medspace = 2^{10} \) (generators) |
Outer automorphisms: | $C_2^2\times C_4$, of order \(16\)\(\medspace = 2^{4} \) |
Composition factors: | $C_2$ x 8 |
Nilpotency class: | $5$ |
Derived length: | $3$ |
This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).
Group statistics
Order | 1 | 2 | 4 | 8 | 16 | |
---|---|---|---|---|---|---|
Elements | 1 | 31 | 64 | 32 | 128 | 256 |
Conjugacy classes | 1 | 6 | 9 | 6 | 12 | 34 |
Divisions | 1 | 6 | 7 | 2 | 2 | 18 |
Autjugacy classes | 1 | 5 | 6 | 2 | 2 | 16 |
Dimension | 1 | 2 | 4 | 32 | |
---|---|---|---|---|---|
Irr. complex chars. | 16 | 4 | 14 | 0 | 34 |
Irr. rational chars. | 4 | 4 | 9 | 1 | 18 |
Minimal Presentations
Permutation degree: | $64$ |
Transitive degree: | $64$ |
Rank: | $2$ |
Inequivalent generating pairs: | $24$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | 4 | 8 | 32 |
Arbitrary | 4 | 8 | 32 |
Constructions
Presentation: | ${\langle a, b, c, d, e \mid b^{4}=c^{2}=d^{4}=e^{2}=[a,d]=[b,d]=[c,d]=[c,e]= \!\cdots\! \rangle}$ | |||||||
Permutation group: | Degree $64$ $\langle(1,40,8,36,4,38,6,34,2,39,7,35,3,37,5,33)(9,48,16,44,12,46,14,42,10,47,15,43,11,45,13,41) \!\cdots\! \rangle$ | |||||||
Direct product: | not isomorphic to a non-trivial direct product | |||||||
Semidirect product: | $(C_2^3.\OD_{16})$ $\,\rtimes\,$ $C_2$ | $(C_2^3.\OD_{16})$ $\,\rtimes\,$ $C_2$ | more information | |||||
Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||
Non-split product: | $C_4$ . $(C_2\wr C_4)$ (2) | $(C_2^2\times C_8)$ . $D_4$ | $(C_4.C_2^4)$ . $C_4$ | $(C_4.C_2^3)$ . $C_8$ (2) | all 17 |
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
Abelianization: | $C_{2} \times C_{8} $ |
Schur multiplier: | $C_{2}^{2}$ |
Commutator length: | $1$ |
Subgroups
There are 587 subgroups in 159 conjugacy classes, 23 normal (21 characteristic).
Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_4$ | $G/Z \simeq$ $C_2\wr C_4$ |
Commutator: | $G' \simeq$ $C_2\times Q_8$ | $G/G' \simeq$ $C_2\times C_8$ |
Frattini: | $\Phi \simeq$ $(C_2^2\times C_8):C_2$ | $G/\Phi \simeq$ $C_2^2$ |
Fitting: | $\operatorname{Fit} \simeq$ $(C_2^2\times C_8).D_4$ | $G/\operatorname{Fit} \simeq$ $C_1$ |
Radical: | $R \simeq$ $(C_2^2\times C_8).D_4$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_2$ | $G/\operatorname{soc} \simeq$ $C_2^4:C_8$ |
2-Sylow subgroup: | $P_{ 2 } \simeq$ $(C_2^2\times C_8).D_4$ |
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.
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Series
Derived series | $(C_2^2\times C_8).D_4$ | $\rhd$ | $C_2\times Q_8$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ | ||||||||||
Chief series | $(C_2^2\times C_8).D_4$ | $\rhd$ | $(C_2^2\times C_8):C_2^2$ | $\rhd$ | $(C_2^2\times C_8):C_2$ | $\rhd$ | $D_4:C_2^2$ | $\rhd$ | $C_2^2\times C_4$ | $\rhd$ | $C_2\times C_4$ | $\rhd$ | $C_2^2$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ |
Lower central series | $(C_2^2\times C_8).D_4$ | $\rhd$ | $C_2\times Q_8$ | $\rhd$ | $C_2\times C_4$ | $\rhd$ | $C_2^2$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ | ||||||
Upper central series | $C_1$ | $\lhd$ | $C_4$ | $\lhd$ | $C_2\times C_4$ | $\lhd$ | $C_2^2\times C_4$ | $\lhd$ | $(C_2^2\times C_8):C_2$ | $\lhd$ | $(C_2^2\times C_8).D_4$ |
Supergroups
Character theory
Complex character table
See the $34 \times 34$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
See the $18 \times 18$ rational character table.