Group information
Description: | $D_{24}:C_8$ |
Order: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Exponent: | \(48\)\(\medspace = 2^{4} \cdot 3 \) |
Automorphism group: | $1536.408134076$, of order \(1536\)\(\medspace = 2^{9} \cdot 3 \) (generators) |
Outer automorphisms: | $C_2^4$, of order \(16\)\(\medspace = 2^{4} \) |
Composition factors: | $C_2$ x 7, $C_3$ |
Derived length: | $2$ |
This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.
Group statistics
Order | 1 | 2 | 3 | 4 | 6 | 8 | 12 | 16 | 24 | |
---|---|---|---|---|---|---|---|---|---|---|
Elements | 1 | 27 | 2 | 36 | 6 | 96 | 24 | 96 | 96 | 384 |
Conjugacy classes | 1 | 3 | 1 | 8 | 3 | 16 | 12 | 4 | 24 | 72 |
Divisions | 1 | 3 | 1 | 5 | 2 | 7 | 4 | 1 | 6 | 30 |
Autjugacy classes | 1 | 3 | 1 | 4 | 2 | 6 | 3 | 1 | 5 | 26 |
Dimension | 1 | 2 | 4 | 8 | 16 | |
---|---|---|---|---|---|---|
Irr. complex chars. | 16 | 44 | 12 | 0 | 0 | 72 |
Irr. rational chars. | 4 | 6 | 10 | 8 | 2 | 30 |
Minimal Presentations
Permutation degree: | $19$ |
Transitive degree: | $48$ |
Rank: | $2$ |
Inequivalent generating pairs: | $24$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | 4 | 8 | 16 |
Arbitrary | 4 | 8 | 10 |
Constructions
Presentation: | $\langle a, b, c \mid a^{2}=b^{24}=c^{8}=[a,c]=1, b^{a}=b^{11}c, c^{b}=c^{5} \rangle$ | |||||||
Permutation group: | Degree $19$ $\langle(1,2,5,10,6,3,13,14,7,12,4,15,8,11,16,9)(18,19), (1,3)(2,8)(4,14)(5,9)(6,12) \!\cdots\! \rangle$ | |||||||
Direct product: | not isomorphic to a non-trivial direct product | |||||||
Semidirect product: | $D_{24}$ $\,\rtimes\,$ $C_8$ | $(C_8:C_8)$ $\,\rtimes\,$ $S_3$ | $C_3$ $\,\rtimes\,$ $(D_8:C_8)$ | $(C_3:Q_{16})$ $\,\rtimes\,$ $C_8$ | all 6 | |||
Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||
Non-split product: | $C_{24}$ . $D_8$ | $C_8$ . $D_{24}$ | $C_{24}$ . $\SD_{16}$ | $C_{24}$ . $\OD_{16}$ | all 29 |
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: | $C_{2}$ |
Commutator length: | $1$ |
Subgroups
There are 326 subgroups in 96 conjugacy classes, 37 normal, and all normal subgroups are characteristic.
Characteristic subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_4$ | $G/Z \simeq$ $D_{12}:C_4$ |
Commutator: | $G' \simeq$ $C_{24}$ | $G/G' \simeq$ $C_2\times C_8$ |
Frattini: | $\Phi \simeq$ $C_4\times C_8$ | $G/\Phi \simeq$ $D_6$ |
Fitting: | $\operatorname{Fit} \simeq$ $C_8:C_{24}$ | $G/\operatorname{Fit} \simeq$ $C_2$ |
Radical: | $R \simeq$ $D_{24}:C_8$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_6$ | $G/\operatorname{soc} \simeq$ $D_4:C_8$ |
2-Sylow subgroup: | $P_{ 2 } \simeq$ $D_8:C_8$ | |
3-Sylow subgroup: | $P_{ 3 } \simeq$ $C_3$ |
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 | $D_{24}:C_8$ | $\rhd$ | $C_{24}$ | $\rhd$ | $C_1$ | ||||||||||||
Chief series | $D_{24}:C_8$ | $\rhd$ | $D_{24}:C_4$ | $\rhd$ | $C_4\times C_{24}$ | $\rhd$ | $C_2\times C_{24}$ | $\rhd$ | $C_{24}$ | $\rhd$ | $C_8$ | $\rhd$ | $C_4$ | $\rhd$ | $C_2$ | $\rhd$ | $C_1$ |
Lower central series | $D_{24}:C_8$ | $\rhd$ | $C_{24}$ | $\rhd$ | $C_{12}$ | $\rhd$ | $C_6$ | $\rhd$ | $C_3$ | ||||||||
Upper central series | $C_1$ | $\lhd$ | $C_4$ | $\lhd$ | $C_2\times C_8$ | $\lhd$ | $C_4\times C_8$ | $\lhd$ | $C_8:C_8$ |
Character theory
Complex character table
See the $72 \times 72$ character table. Alternatively, you may search for characters of this group with desired properties.
Rational character table
See the $30 \times 30$ rational character table.