Abstract group 3993.6: $C_{11}\wr C_3$

• Label: 3993.6
• Order: \( 3 \cdot 11^{3} \)
• Exponent: \( 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 3 \cdot 11 \)
• $\card{Z(G)}$: \( 11 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \cdot 5^{2} \)
• Perm deg.: $33$
• Trans deg.: $33$
• Rank: $2$

Group information

Description:	$C_{11}\wr C_3$
Order:	\(3993\)\(\medspace = 3 \cdot 11^{3} \)
Exponent:	\(33\)\(\medspace = 3 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{60}.C_5.C_2^3$, of order \(290400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_3$, $C_{11}$ x 3
Derived length:	$2$

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

Group statistics

Order	1	3	11	33
Elements	1	242	1330	2420	3993
Conjugacy classes	1	2	450	20	473
Divisions	1	1	45	1	48
Autjugacy classes	1	1	3	1	6

Dimension	1	2	3	10	20	30
Irr. complex chars.	33	0	440	0	0	0	473
Irr. rational chars.	1	1	0	1	1	44	48

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$33$
Rank:	$2$
Inequivalent generating pairs:	$48$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	3	6	30
Arbitrary	3	6	30

Constructions

Groups of Lie type:	$\ASigmaL(1,1331)$
Presentation:	$\langle a, b, c \mid a^{33}=b^{11}=c^{11}=[b,c]=1, b^{a}=b^{9}c, c^{a}=b^{8}c \rangle$
Permutation group:	Degree $33$ $\langle(1,23,12)(2,24,13)(3,25,14)(4,26,15)(5,27,16)(6,28,17)(7,29,18)(8,30,19) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 56 & 66 \\ 55 & 67 \end{array}\right), \left(\begin{array}{rr} 56 & 33 \\ 110 & 67 \end{array}\right), \left(\begin{array}{rr} 120 & 1 \\ 120 & 0 \end{array}\right), \left(\begin{array}{rr} 1 & 0 \\ 11 & 1 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/121\Z)$
Transitive group:	33T23				more information
Direct product:	$C_{11}$ $\, \times\, $ $(C_{11}^2:C_3)$
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $C_3$	$C_{11}^2$ $\,\rtimes\,$ $C_{33}$			more information
Trans. wreath product:	$C_{11}$$\ \wr\ $$C_3$

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

Homology

Abelianization:	$C_{33} \simeq C_{3} \times C_{11}$
Schur multiplier:	$C_{11}$
Commutator length:	$1$

Subgroups

There are 512 subgroups in 96 conjugacy classes, 6 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

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

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

Click on a subgroup in the diagram to see information about it.

Classes of subgroups up to conjugation

Order 3993: $C_{11}\wr C_3$

Order 1331: $C_{11}^3$

Order 363: $C_{11}^2:C_3$

Order 121: $C_{11}^2$ x 45

Order 33: $C_{33}$

Order 11: $C_{11}$ x 45

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 3993: $C_{11}\wr C_3$

Order 1331: $C_{11}^3$

Order 363: $C_{11}^2:C_3$

Order 121: $C_{11}^2$ x 3

Order 33: $C_{33}$

Order 11: $C_{11}$ x 3

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 3993: $C_{11}\wr C_3$ ($C_1$)

Order 1331: $C_{11}^3$ ($C_3$)

Order 363: $C_{11}^2:C_3$ ($C_{11}$)

Order 121: $C_{11}^2$ ($C_{33}$)

Order 11: $C_{11}$ ($C_{11}^2:C_3$)

Order 1: $C_1$ ($C_{11}\wr C_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 3993: $C_{11}\wr C_3$ ($C_1$)

Order 1331: $C_{11}^3$ ($C_3$)

Order 363: $C_{11}^2:C_3$ ($C_{11}$)

Order 121: $C_{11}^2$ ($C_{33}$)

Order 11: $C_{11}$ ($C_{11}^2:C_3$)

Order 1: $C_1$ ($C_{11}\wr C_3$)

Series

Derived series	$C_{11}\wr C_3$	$\rhd$	$C_{11}^2$	$\rhd$	$C_1$
Chief series	$C_{11}\wr C_3$	$\rhd$	$C_{11}^3$	$\rhd$	$C_{11}^2$	$\rhd$	$C_1$
Lower central series	$C_{11}\wr C_3$	$\rhd$	$C_{11}^2$
Upper central series	$C_1$	$\lhd$	$C_{11}$

Supergroups

This group is a maximal subgroup of 13 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $473 \times 473$ 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 $48 \times 48$ rational character table.