Group information
Description: | $C_7:D_{36}$ |
Order: | \(504\)\(\medspace = 2^{3} \cdot 3^{2} \cdot 7 \) |
Exponent: | \(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \) |
Automorphism group: | Group of order \(9072\)\(\medspace = 2^{4} \cdot 3^{4} \cdot 7 \) (generators) |
Outer automorphisms: | $C_6^2$, of order \(36\)\(\medspace = 2^{2} \cdot 3^{2} \) |
Composition factors: | $C_2$ x 3, $C_3$ x 2, $C_7$ |
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 | 7 | 9 | 12 | 14 | 18 | 21 | 36 | 42 | 63 | 126 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Elements | 1 | 145 | 2 | 14 | 2 | 6 | 6 | 28 | 114 | 6 | 12 | 84 | 12 | 36 | 36 | 504 |
Conjugacy classes | 1 | 3 | 1 | 1 | 1 | 3 | 3 | 2 | 9 | 3 | 3 | 6 | 3 | 9 | 9 | 57 |
Divisions | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 18 |
Autjugacy classes | 1 | 3 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 18 |
Dimension | 1 | 2 | 4 | 6 | 12 | 36 | |
---|---|---|---|---|---|---|---|
Irr. complex chars. | 4 | 29 | 24 | 0 | 0 | 0 | 57 |
Irr. rational chars. | 4 | 3 | 1 | 4 | 4 | 2 | 18 |
Minimal Presentations
Permutation degree: | $20$ |
Transitive degree: | $252$ |
Rank: | $2$ |
Inequivalent generating pairs: | $6$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | 4 | 4 | 72 |
Arbitrary | 4 | 4 | 14 |
Constructions
Presentation: | $\langle a, b, c \mid a^{2}=b^{36}=c^{7}=[a,c]=1, b^{a}=b^{35}, c^{b}=c^{6} \rangle$ | |||||||
Permutation group: | Degree $20$ $\langle(2,3)(4,5)(6,7)(8,9,10,11), (9,11)(13,15)(14,18)(16,17)(19,20), (8,10)(9,11) \!\cdots\! \rangle$ | |||||||
Direct product: | not isomorphic to a non-trivial direct product | |||||||
Semidirect product: | $D_{18}$ $\,\rtimes\,$ $D_7$ | $C_{63}$ $\,\rtimes\,$ $D_4$ | $C_7$ $\,\rtimes\,$ $D_{36}$ | $D_{126}$ $\,\rtimes\,$ $C_2$ | all 8 | |||
Trans. wreath product: | not isomorphic to a non-trivial transitive wreath product | |||||||
Non-split product: | $C_{42}$ . $D_6$ | $C_{21}$ . $D_{12}$ | $C_{18}$ . $D_{14}$ | $C_{14}$ . $D_{18}$ | all 9 |
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
Abelianization: | $C_{2}^{2} $ |
Schur multiplier: | $C_{2}$ |
Commutator length: | $1$ |
Subgroups
There are 530 subgroups in 48 conjugacy classes, 19 normal, and all normal subgroups are characteristic.
Characteristic subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_2$ | $G/Z \simeq$ $D_7\times D_9$ |
Commutator: | $G' \simeq$ $C_{126}$ | $G/G' \simeq$ $C_2^2$ |
Frattini: | $\Phi \simeq$ $C_6$ | $G/\Phi \simeq$ $S_3\times D_7$ |
Fitting: | $\operatorname{Fit} \simeq$ $C_{126}$ | $G/\operatorname{Fit} \simeq$ $C_2^2$ |
Radical: | $R \simeq$ $C_7:D_{36}$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_{42}$ | $G/\operatorname{soc} \simeq$ $D_6$ |
2-Sylow subgroup: | $P_{ 2 } \simeq$ $D_4$ | |
3-Sylow subgroup: | $P_{ 3 } \simeq$ $C_9$ | |
7-Sylow subgroup: | $P_{ 7 } \simeq$ $C_7$ |
Subgroup diagram and profile
For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
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Subgroup information
Click on a subgroup in the diagram to see information about it.
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Series
Derived series | $C_7:D_{36}$ | $\rhd$ | $C_{126}$ | $\rhd$ | $C_1$ | ||||||||
Chief series | $C_7:D_{36}$ | $\rhd$ | $C_7\times D_{18}$ | $\rhd$ | $C_{126}$ | $\rhd$ | $C_{63}$ | $\rhd$ | $C_{21}$ | $\rhd$ | $C_7$ | $\rhd$ | $C_1$ |
Lower central series | $C_7:D_{36}$ | $\rhd$ | $C_{126}$ | $\rhd$ | $C_{63}$ | ||||||||
Upper central series | $C_1$ | $\lhd$ | $C_2$ |
Supergroups
This group is a maximal subgroup of 17 larger groups in the database.
This group is a maximal quotient of 13 larger groups in the database.
Character theory
Complex character table
See the $57 \times 57$ 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.