Abstract group 1210.27: $C_{11}\times C_{110}$

• Label: 1210.27
• Order: \( 2 \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Abelian: yes
• $\card{\Aut(G)}$: \( 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 \)
• Perm deg.: $29$
• Trans deg.: $1210$
• Rank: $2$

Group information

Description:	$C_{11}\times C_{110}$
Order:	\(1210\)\(\medspace = 2 \cdot 5 \cdot 11^{2} \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$C_4\times C_{10}.\PSL(2,11).C_2$, of order \(52800\)\(\medspace = 2^{6} \cdot 3 \cdot 5^{2} \cdot 11 \)
Composition factors:	$C_2$, $C_5$, $C_{11}$ 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	2	5	10	11	22	55	110
Elements	1	1	4	4	120	120	480	480	1210
Conjugacy classes	1	1	4	4	120	120	480	480	1210
Divisions	1	1	1	1	12	12	12	12	52
Autjugacy classes	1	1	1	1	1	1	1	1	8

Dimension	1	4	10	40
Irr. complex chars.	1210	0	0	0	1210
Irr. rational chars.	2	2	24	24	52

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$1210$
Rank:	$2$
Inequivalent generating pairs:	$18$

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 $\langle a, b \mid a^{11}=b^{110}=1 \rangle$
Permutation group:	Degree $29$ $\langle(1,2), (3,7,6,5,4), (8,18,17,16,15,14,13,12,11,10,9), (19,29,28,27,26,25,24,23,22,21,20)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 27 & 0 \\ 0 & 282 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 0 & 270 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{331})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \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

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

Homology

Primary decomposition:	$C_{2} \times C_{5} \times C_{11}^{2}$
Schur multiplier:	$C_{11}$
Commutator length:	$0$

Subgroups

There are 56 subgroups, all normal (8 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_{11}\times C_{110}$	$G/Z \simeq$ $C_1$
Commutator:	$G' \simeq$ $C_1$	$G/G' \simeq$ $C_{11}\times C_{110}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}\times C_{110}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_{11}\times C_{110}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{110}$	$G/\operatorname{soc} \simeq$ $C_1$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
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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Subgroup information

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Classes of subgroups up to conjugation

Order 1210: $C_{11}\times C_{110}$

Order 605: $C_{11}\times C_{55}$

Order 242: $C_{11}\times C_{22}$

Order 121: $C_{11}^2$

Order 110: $C_{110}$ x 12

Order 55: $C_{55}$ x 12

Order 22: $C_{22}$ x 12

Order 11: $C_{11}$ x 12

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1210: $C_{11}\times C_{110}$

Order 605: $C_{11}\times C_{55}$

Order 242: $C_{11}\times C_{22}$

Order 121: $C_{11}^2$

Order 110: $C_{110}$

Order 55: $C_{55}$

Order 22: $C_{22}$

Order 11: $C_{11}$

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1210: $C_{11}\times C_{110}$ ($C_1$)

Order 605: $C_{11}\times C_{55}$ ($C_2$)

Order 242: $C_{11}\times C_{22}$ ($C_5$)

Order 121: $C_{11}^2$ ($C_{10}$)

Order 110: $C_{110}$ ($C_{11}$) x 12

Order 55: $C_{55}$ ($C_{22}$) x 12

Order 22: $C_{22}$ ($C_{55}$) x 12

Order 11: $C_{11}$ ($C_{110}$) x 12

Order 10: $C_{10}$ ($C_{11}^2$)

Order 5: $C_5$ ($C_{11}\times C_{22}$)

Order 2: $C_2$ ($C_{11}\times C_{55}$)

Order 1: $C_1$ ($C_{11}\times C_{110}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1210: $C_{11}\times C_{110}$ ($C_1$)

Order 605: $C_{11}\times C_{55}$ ($C_2$)

Order 242: $C_{11}\times C_{22}$ ($C_5$)

Order 121: $C_{11}^2$ ($C_{10}$)

Order 110: $C_{110}$ ($C_{11}$)

Order 55: $C_{55}$ ($C_{22}$)

Order 22: $C_{22}$ ($C_{55}$)

Order 11: $C_{11}$ ($C_{110}$)

Order 10: $C_{10}$ ($C_{11}^2$)

Order 5: $C_5$ ($C_{11}\times C_{22}$)

Order 2: $C_2$ ($C_{11}\times C_{55}$)

Order 1: $C_1$ ($C_{11}\times C_{110}$)

Series

Derived series	$C_{11}\times C_{110}$	$\rhd$	$C_1$
Chief series	$C_{11}\times C_{110}$	$\rhd$	$C_{110}$	$\rhd$	$C_{55}$	$\rhd$	$C_{11}$	$\rhd$	$C_1$
Lower central series	$C_{11}\times C_{110}$	$\rhd$	$C_1$
Upper central series	$C_1$	$\lhd$	$C_{11}\times C_{110}$

Supergroups

This group is a maximal subgroup of 26 larger groups in the database.

This group is a maximal quotient of 15 larger groups in the database.

Character theory

Complex character table

The $1210 \times 1210$ character table is not available for this group.

Rational character table

See the $52 \times 52$ rational character table.