Group information
Description: | $C_{199}$ |
Order: | \(199\) |
Exponent: | \(199\) |
Automorphism group: | $C_{198}$, of order \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) (generators) |
Outer automorphisms: | $C_{198}$, of order \(198\)\(\medspace = 2 \cdot 3^{2} \cdot 11 \) |
Composition factors: | $C_{199}$ |
Nilpotency class: | $1$ |
Derived length: | $1$ |
This group is cyclic (hence abelian, nilpotent, solvable, supersolvable, monomial, elementary, hyperelementary, metacyclic, metabelian, a Z-group, and an A-group), a $p$-group, and simple.
Group statistics
Order | 1 | 199 | |
---|---|---|---|
Elements | 1 | 198 | 199 |
Conjugacy classes | 1 | 198 | 199 |
Divisions | 1 | 1 | 2 |
Autjugacy classes | 1 | 1 | 2 |
Dimension | 1 | 198 | |
---|---|---|---|
Irr. complex chars. | 199 | 0 | 199 |
Irr. rational chars. | 1 | 1 | 2 |
Minimal Presentations
Permutation degree: | $199$ |
Transitive degree: | $199$ |
Rank: | $1$ |
Inequivalent generators: | $1$ |
Minimal degrees of faithful linear representations
Over $\mathbb{C}$ | Over $\mathbb{R}$ | Over $\mathbb{Q}$ | |
---|---|---|---|
Irreducible | 1 | 2 | 198 |
Arbitrary | 1 | 2 | 198 |
Constructions
Presentation: | $\langle a \mid a^{199}=1 \rangle$ | |||||||||
Permutation group: | Degree $199$ $\langle(1,199,198,197,196,195,194,193,192,191,190,189,188,187,186,185,184,183,182,181,180,179,178,177,176,175,174,173,172,171,170,169,168,167,166,165,164,163,162,161,160,159,158,157,156,155,154,153,152,151,150,149,148,147,146,145,144,143,142,141,140,139,138,137,136,135,134,133,132,131,130,129,128,127,126,125,124,123,122,121,120,119,118,117,116,115,114,113,112,111,110,109,108,107,106,105,104,103,102,101,100,99,98,97,96,95,94,93,92,91,90,89,88,87,86,85,84,83,82,81,80,79,78,77,76,75,74,73,72,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2) \!\cdots\! \rangle$ | |||||||||
Matrix group: | $\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$ | |||||||||
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 |
Elements of the group are displayed as words in the generators from the presentation given above.
Homology
Primary decomposition: | $C_{199}$ |
Schur multiplier: | $C_1$ |
Commutator length: | $0$ |
Subgroups
There are 2 subgroups, all normal, and all normal subgroups are characteristic.
Characteristic subgroups are shown in this color.
Special subgroups
Center: | $Z \simeq$ $C_{199}$ | $G/Z \simeq$ $C_1$ |
Commutator: | $G' \simeq$ $C_1$ | $G/G' \simeq$ $C_{199}$ |
Frattini: | $\Phi \simeq$ $C_1$ | $G/\Phi \simeq$ $C_{199}$ |
Fitting: | $\operatorname{Fit} \simeq$ $C_{199}$ | $G/\operatorname{Fit} \simeq$ $C_1$ |
Radical: | $R \simeq$ $C_{199}$ | $G/R \simeq$ $C_1$ |
Socle: | $\operatorname{soc} \simeq$ $C_{199}$ | $G/\operatorname{soc} \simeq$ $C_1$ |
199-Sylow subgroup: | $P_{ 199 } \simeq$ $C_{199}$ |
Subgroup diagram and profile
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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 | $C_{199}$ | $\rhd$ | $C_1$ |
Chief series | $C_{199}$ | $\rhd$ | $C_1$ |
Lower central series | $C_{199}$ | $\rhd$ | $C_1$ |
Upper central series | $C_1$ | $\lhd$ | $C_{199}$ |
Supergroups
This group is a maximal subgroup of 8 larger groups in the database.
This group is a maximal quotient of 5 larger groups in the database.
Character theory
Complex character table
See the $199 \times 199$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.
Rational character table
1A | 199A | ||
Size | 1 | 198 | |
199 P | 1A | 199A | |
199.1.1a | |||
199.1.1b |