Abstract group 146410.q: $C_{11}^3:C_{110}$

• Label: 146410.q
• Order: \( 2 \cdot 5 \cdot 11^{4} \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \cdot 11 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{2} \cdot 5^{2} \cdot 11^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \cdot 5 \cdot 11 \)
• Perm deg.: $123$
• Trans deg.: $242$
• Rank: $2$

Group information

Description:	$C_{11}^3:C_{110}$
Order:	\(146410\)\(\medspace = 2 \cdot 5 \cdot 11^{4} \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_{11}^2.C_{10}^2$, of order \(16105100\)\(\medspace = 2^{2} \cdot 5^{2} \cdot 11^{5} \)
Composition factors:	$C_2$, $C_5$, $C_{11}$ x 4
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), and metabelian.

Group statistics

Order	1	2	5	10	11	22	55	110
Elements	1	1	5324	5324	14640	14640	53240	53240	146410
Conjugacy classes	1	1	4	4	56	56	40	40	202
Divisions	1	1	1	1	16	16	1	1	38
Autjugacy classes	1	1	4	4	5	5	4	4	28

Dimension	1	4	5	10	40	50	55	110
Irr. complex chars.	110	0	44	0	0	0	48	0	202
Irr. rational chars.	2	2	0	4	2	4	0	24	38

Minimal presentations

Permutation degree:	$123$
Transitive degree:	$242$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	55	110	110
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d \mid a^{110}=b^{11}=c^{11}=d^{11}=[b,c]=[b,d]=[c,d]= \!\cdots\! \rangle}$
Permutation group:	Degree $123$ $\langle(1,2,12,61,86,63,27,57,10,56,59,11,29,35,5,24,99,21,98,108,32,41,6,39,37,113,96,53,73,79,85,17,77,44,105,66,80,15,3,19,31,64,36,9,30,4,26,62,33,71,82,16,81,47,7) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 8 & 0 & 1 & 1 \\ 8 & 0 & 8 & 1 \\ 8 & 3 & 9 & 0 \\ 10 & 8 & 3 & 1 \end{array}\right), \left(\begin{array}{rrrr} 0 & 8 & 5 & 0 \\ 1 & 4 & 8 & 5 \\ 7 & 8 & 5 & 3 \\ 3 & 7 & 10 & 9 \end{array}\right), \left(\begin{array}{rrrr} 0 & 10 & 7 & 4 \\ 0 & 1 & 4 & 6 \\ 9 & 4 & 2 & 4 \\ 9 & 10 & 6 & 6 \end{array}\right), \left(\begin{array}{rrrr} 0 & 7 & 4 & 6 \\ 9 & 3 & 0 & 8 \\ 10 & 3 & 4 & 4 \\ 9 & 9 & 2 & 0 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $(C_{11}^3:C_{55})$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $C_{110}$	$(C_{11}^3:C_{22})$ $\,\rtimes\,$ $C_5$	$(C_{11}^3:C_5)$ $\,\rtimes\,$ $C_{22}$	$(C_{11}^3:C_{11})$ $\,\rtimes\,$ $C_{10}$	all 8
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{22}$ . $(C_{11}^2:C_{55})$	$C_{11}^2$ . $(C_{11}:C_{110})$	$C_{11}$ . $(C_{11}^2:C_{110})$	$(C_{11}\times C_{22})$ . $(C_{11}:C_{55})$	more information

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

Homology

Abelianization:	$C_{110} \simeq C_{2} \times C_{5} \times C_{11}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 44168 subgroups in 140 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{11}^3:C_{55}$
Commutator:	$G' \simeq$ $C_{11}^3$	$G/G' \simeq$ $C_{110}$
Frattini:	$\Phi \simeq$ $C_{11}^2$	$G/\Phi \simeq$ $C_{11}:C_{110}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^3:C_{22}$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_{11}^3:C_{110}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{22}$	$G/\operatorname{soc} \simeq$ $C_{11}^2:C_{55}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3:C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 146410: $C_{11}^3:C_{110}$

Order 73205: $C_{11}^3:C_{55}$

Order 29282: $C_{11}^3:C_{22}$

Order 14641: $C_{11}^3:C_{11}$

Order 13310: $C_{11}^3:C_{10}$, $C_{11}^2:C_{110}$

Order 6655: $C_{11}^2:C_{55}$, $C_{11}^3:C_5$

Order 2662: $C_2\times \He_{11}$ x 3, $C_{11}^2\times C_{22}$

Order 1331: $\He_{11}$ x 3, $C_{11}^3$

Order 1210: $C_{11}^2:C_{10}$ x 13, $C_{11}:C_{110}$

Order 605: $C_{11}^2:C_5$ x 13, $C_{11}:C_{55}$

Order 242: $C_{11}\times C_{22}$ x 16

Order 121: $C_{11}^2$ x 16

Order 110: $C_{11}:C_{10}$ x 13, $C_{110}$

Order 55: $C_{11}:C_5$ x 13, $C_{55}$

Order 22: $C_{22}$ x 16

Order 11: $C_{11}$ x 16

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 146410: $C_{11}^3:C_{110}$

Order 73205: $C_{11}^3:C_{55}$

Order 29282: $C_{11}^3:C_{22}$

Order 14641: $C_{11}^3:C_{11}$

Order 13310: $C_{11}^3:C_{10}$, $C_{11}^2:C_{110}$

Order 6655: $C_{11}^2:C_{55}$, $C_{11}^3:C_5$

Order 2662: $C_2\times \He_{11}$ x 2, $C_{11}^2\times C_{22}$

Order 1331: $\He_{11}$ x 2, $C_{11}^3$

Order 1210: $C_{11}^2:C_{10}$ x 3, $C_{11}:C_{110}$

Order 605: $C_{11}^2:C_5$ x 3, $C_{11}:C_{55}$

Order 242: $C_{11}\times C_{22}$ x 5

Order 121: $C_{11}^2$ x 5

Order 110: $C_{11}:C_{10}$ x 3, $C_{110}$

Order 55: $C_{11}:C_5$ x 3, $C_{55}$

Order 22: $C_{22}$ x 5

Order 11: $C_{11}$ x 5

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 146410: $C_{11}^3:C_{110}$ ($C_1$)

Order 73205: $C_{11}^3:C_{55}$ ($C_2$)

Order 29282: $C_{11}^3:C_{22}$ ($C_5$)

Order 14641: $C_{11}^3:C_{11}$ ($C_{10}$)

Order 13310: $C_{11}^3:C_{10}$ ($C_{11}$)

Order 6655: $C_{11}^3:C_5$ ($C_{22}$)

Order 2662: $C_2\times \He_{11}$ ($C_{11}:C_5$), $C_{11}^2\times C_{22}$ ($C_{55}$)

Order 1331: $C_{11}^3$ ($C_{110}$), $\He_{11}$ ($C_{11}:C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{11}:C_{55}$)

Order 121: $C_{11}^2$ ($C_{11}:C_{110}$)

Order 22: $C_{22}$ ($C_{11}^2:C_{55}$)

Order 11: $C_{11}$ ($C_{11}^2:C_{110}$)

Order 2: $C_2$ ($C_{11}^3:C_{55}$)

Order 1: $C_1$ ($C_{11}^3:C_{110}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 146410: $C_{11}^3:C_{110}$ ($C_1$)

Order 73205: $C_{11}^3:C_{55}$ ($C_2$)

Order 29282: $C_{11}^3:C_{22}$ ($C_5$)

Order 14641: $C_{11}^3:C_{11}$ ($C_{10}$)

Order 13310: $C_{11}^3:C_{10}$ ($C_{11}$)

Order 6655: $C_{11}^3:C_5$ ($C_{22}$)

Order 2662: $C_2\times \He_{11}$ ($C_{11}:C_5$), $C_{11}^2\times C_{22}$ ($C_{55}$)

Order 1331: $C_{11}^3$ ($C_{110}$), $\He_{11}$ ($C_{11}:C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{11}:C_{55}$)

Order 121: $C_{11}^2$ ($C_{11}:C_{110}$)

Order 22: $C_{22}$ ($C_{11}^2:C_{55}$)

Order 11: $C_{11}$ ($C_{11}^2:C_{110}$)

Order 2: $C_2$ ($C_{11}^3:C_{55}$)

Order 1: $C_1$ ($C_{11}^3:C_{110}$)

Series

Derived series

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3:C_{55}$

$\rhd$

$C_{11}^3:C_{55}$

$\rhd$

$C_{11}^3:C_{11}$

$\rhd$

$C_{11}^3:C_{11}$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3:C_{110}$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 2 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $202 \times 202$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $38 \times 38$ rational character table.