Abstract group 1089.4: $C_{11}\times C_{99}$

• Label: 1089.4
• Order: \( 3^{2} \cdot 11^{2} \)
• Exponent: \( 3^{2} \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
• Perm deg.: $31$
• Trans deg.: $1089$
• Rank: $2$

Group information

Description:	$C_{11}\times C_{99}$
Order:	\(1089\)\(\medspace = 3^{2} \cdot 11^{2} \)
Exponent:	\(99\)\(\medspace = 3^{2} \cdot 11 \)
Automorphism group:	$(C_2\times C_{30}).\PSL(2,11).C_2$, of order \(79200\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Composition factors:	$C_3$ x 2, $C_{11}$ x 2
Nilpotency class:	$1$
Derived length:	$1$

This group is abelian (hence nilpotent, solvable, supersolvable, monomial, metabelian, and an A-group), elementary for $p = 11$ (hence hyperelementary), and metacyclic.

Group statistics

Order	1	3	9	11	33	99
Elements	1	2	6	120	240	720	1089
Conjugacy classes	1	2	6	120	240	720	1089
Divisions	1	1	1	12	12	12	39
Autjugacy classes	1	1	1	1	1	1	6

Dimension	1	2	6	10	20	60
Irr. complex chars.	1089	0	0	0	0	0	1089
Irr. rational chars.	1	1	1	12	12	12	39

Minimal presentations

Permutation degree:	$31$
Transitive degree:	$1089$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	2	not computed	not computed

Constructions

Presentation:	Abelian group $\langle a, b \mid a^{11}=b^{99}=1 \rangle$
Permutation group:	Degree $31$ $\langle(1,9,6,3,8,5,2,7,4), (1,3,2)(4,6,5)(7,9,8), (10,20,19,18,17,16,15,14,13,12,11), (21,31,30,29,28,27,26,25,24,23,22)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 9 & 0 \\ 0 & 177 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 125 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{199})$
Direct product:	$C_9$ $\, \times\, $ $C_{11}$ ${}^2$
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:	$C_{33}$ . $C_{33}$ (12)	$(C_{11}\times C_{33})$ . $C_3$	$C_3$ . $(C_{11}\times C_{33})$		more information

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Primary decomposition:	$C_{9} \times C_{11}^{2}$
Schur multiplier:	$C_{11}$
Commutator length:	$0$

Subgroups

There are 42 subgroups, all normal (6 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{11}\times C_{99}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{11}\times C_{99}$
Frattini:	$\Phi \simeq$ $C_3$	$G/\Phi \simeq$ $C_{11}\times C_{33}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{99}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{11}\times C_{99}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{33}$	$G/\operatorname{soc} \simeq$ $C_3$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_9$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^2$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1089: $C_{11}\times C_{99}$

Order 363: $C_{11}\times C_{33}$

Order 121: $C_{11}^2$

Order 99: $C_{99}$ x 12

Order 33: $C_{33}$ x 12

Order 11: $C_{11}$ x 12

Order 9: $C_9$

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1089: $C_{11}\times C_{99}$

Order 363: $C_{11}\times C_{33}$

Order 121: $C_{11}^2$

Order 99: $C_{99}$

Order 33: $C_{33}$

Order 11: $C_{11}$

Order 9: $C_9$

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1089: $C_{11}\times C_{99}$ ($C_1$)

Order 363: $C_{11}\times C_{33}$ ($C_3$)

Order 121: $C_{11}^2$ ($C_9$)

Order 99: $C_{99}$ ($C_{11}$) x 12

Order 33: $C_{33}$ ($C_{33}$) x 12

Order 11: $C_{11}$ ($C_{99}$) x 12

Order 9: $C_9$ ($C_{11}^2$)

Order 3: $C_3$ ($C_{11}\times C_{33}$)

Order 1: $C_1$ ($C_{11}\times C_{99}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1089: $C_{11}\times C_{99}$ ($C_1$)

Order 363: $C_{11}\times C_{33}$ ($C_3$)

Order 121: $C_{11}^2$ ($C_9$)

Order 99: $C_{99}$ ($C_{11}$)

Order 33: $C_{33}$ ($C_{33}$)

Order 11: $C_{11}$ ($C_{99}$)

Order 9: $C_9$ ($C_{11}^2$)

Order 3: $C_3$ ($C_{11}\times C_{33}$)

Order 1: $C_1$ ($C_{11}\times C_{99}$)

Series

Derived series	$C_{11}\times C_{99}$	$\rhd$	$C_1$
Chief series	$C_{11}\times C_{99}$	$\rhd$	$C_{99}$	$\rhd$	$C_{33}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$C_{11}\times C_{99}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{11}\times C_{99}$

Supergroups

This group is a maximal subgroup of 10 larger groups in the database.

This group is a maximal quotient of 8 larger groups in the database.

Character theory

Complex character table

The $1089 \times 1089$ character table is not available for this group.

Rational character table

See the $39 \times 39$ rational character table.