Abstract group 13310.u: $C_{11}^3:C_{10}$

• Label: 13310.u
• Order: \( 2 \cdot 5 \cdot 11^{3} \)
• Exponent: \( 2 \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{6} \cdot 19 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11^{3} \cdot 19 \)
• Perm deg.: $35$
• Trans deg.: $2662$
• Rank: $4$

Group information

Description:	$C_{11}^3:C_{10}$
Order:	\(13310\)\(\medspace = 2 \cdot 5 \cdot 11^{3} \)
Exponent:	\(110\)\(\medspace = 2 \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_{10}.\PSL(3,11)$, of order \(2827411356000\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{3} \cdot 7 \cdot 11^{6} \cdot 19 \)
Composition factors:	$C_2$, $C_5$, $C_{11}$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	5	10	11	22
Elements	1	1	5324	5324	1330	1330	13310
Conjugacy classes	1	1	4	4	266	266	542
Divisions	1	1	1	1	133	133	270
Autjugacy classes	1	1	4	4	1	1	12

Dimension	1	4	5	10
Irr. complex chars.	10	0	532	0	542
Irr. rational chars.	2	2	0	266	270

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$2662$
Rank:	$4$
Inequivalent generating quadruples:	$9360$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{5}=b^{11}=c^{11}=d^{22}=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{9}, c^{a}=c^{9}, d^{a}=d^{9} \rangle$
Permutation group:	Degree $35$ $\langle(1,2,4,8,10)(5,6,7,11,9)(12,13,15,19,20)(14,17,16,18,22)(23,24,27,28,33) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 5 & 7 & 10 & 8 \\ 6 & 5 & 10 & 10 \\ 3 & 9 & 4 & 4 \\ 3 & 3 & 5 & 4 \end{array}\right), \left(\begin{array}{rrrr} 0 & 4 & 6 & 3 \\ 5 & 0 & 1 & 6 \\ 3 & 2 & 8 & 7 \\ 8 & 3 & 6 & 8 \end{array}\right), \left(\begin{array}{rrrr} 1 & 5 & 7 & 1 \\ 2 & 1 & 7 & 7 \\ 1 & 8 & 8 & 6 \\ 1 & 1 & 9 & 8 \end{array}\right), \left(\begin{array}{rrrr} 6 & 9 & 1 & 4 \\ 4 & 6 & 3 & 1 \\ 8 & 10 & 7 & 2 \\ 2 & 8 & 7 & 7 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	$C_2$ $\, \times\, $ $(C_{11}^3:C_5)$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $C_{10}$	$(C_{11}^2\times C_{22})$ $\,\rtimes\,$ $C_5$	$C_{22}$ $\,\rtimes\,$ $(C_{11}^2:C_5)$	$C_{11}^2$ $\,\rtimes\,$ $(C_{11}:C_{10})$	all 6
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

Abelianization:	$C_{10} \simeq C_{2} \times C_{5}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 38312 subgroups in 1072 conjugacy classes, 538 normal (6 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_{11}^3:C_5$
Commutator:	$G' \simeq$ $C_{11}^3$	$G/G' \simeq$ $C_{10}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}^3:C_{10}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^2\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_5$
Radical:	$R \simeq$ $C_{11}^3:C_{10}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}^2\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_5$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

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

Order 6655: $C_{11}^3:C_5$

Order 2662: $C_{11}^2\times C_{22}$

Order 1331: $C_{11}^3$

Order 1210: $C_{11}^2:C_{10}$ x 133

Order 605: $C_{11}^2:C_5$ x 133

Order 242: $C_{11}\times C_{22}$ x 133

Order 121: $C_{11}^2$ x 133

Order 110: $C_{11}:C_{10}$ x 133

Order 55: $C_{11}:C_5$ x 133

Order 22: $C_{22}$ x 133

Order 11: $C_{11}$ x 133

Order 10: $C_{10}$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

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

Order 6655: $C_{11}^3:C_5$

Order 2662: $C_{11}^2\times C_{22}$

Order 1331: $C_{11}^3$

Order 1210: $C_{11}^2:C_{10}$

Order 605: $C_{11}^2:C_5$

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

Order 121: $C_{11}^2$

Order 110: $C_{11}:C_{10}$

Order 55: $C_{11}:C_5$

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 13310: $C_{11}^3:C_{10}$ ($C_1$)

Order 6655: $C_{11}^3:C_5$ ($C_2$)

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

Order 1331: $C_{11}^3$ ($C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_{11}:C_5$) x 133

Order 121: $C_{11}^2$ ($C_{11}:C_{10}$) x 133

Order 22: $C_{22}$ ($C_{11}^2:C_5$) x 133

Order 11: $C_{11}$ ($C_{11}^2:C_{10}$) x 133

Order 2: $C_2$ ($C_{11}^3:C_5$)

Order 1: $C_1$ ($C_{11}^3:C_{10}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 13310: $C_{11}^3:C_{10}$ ($C_1$)

Order 6655: $C_{11}^3:C_5$ ($C_2$)

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

Order 1331: $C_{11}^3$ ($C_{10}$)

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

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

Order 22: $C_{22}$ ($C_{11}^2:C_5$)

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

Order 2: $C_2$ ($C_{11}^3:C_5$)

Order 1: $C_1$ ($C_{11}^3:C_{10}$)

Series

Derived series

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^2\times C_{22}$

$\rhd$

$C_{11}^2\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{11}\times C_{22}$

$\rhd$

$C_{22}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^3:C_{10}$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 9 larger groups in the database.

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

Character theory

Complex character table

The $542 \times 542$ character table is not available for this group.

Rational character table

See the $270 \times 270$ rational character table (warning: may be slow to load).