Abstract group 7986.g: $C_{11}^3:C_6$

• Label: 7986.g
• Order: \( 2 \cdot 3 \cdot 11^{3} \)
• Exponent: \( 2 \cdot 3 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5^{2} \)
• Perm deg.: $33$
• Trans deg.: $33$
• Rank: $2$

Group information

Description:	$C_{11}^3:C_6$
Order:	\(7986\)\(\medspace = 2 \cdot 3 \cdot 11^{3} \)
Exponent:	\(66\)\(\medspace = 2 \cdot 3 \cdot 11 \)
Automorphism group:	$C_{11}^3.C_{60}.C_5.C_2^3$, of order \(3194400\)\(\medspace = 2^{5} \cdot 3 \cdot 5^{2} \cdot 11^{3} \)
Composition factors:	$C_2$, $C_3$, $C_{11}$ x 3
Derived length:	$2$

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

Group statistics

Order	1	2	3	6	11	33
Elements	1	1331	242	2662	1330	2420	7986
Conjugacy classes	1	1	2	2	225	10	241
Divisions	1	1	1	1	45	1	50
Autjugacy classes	1	1	1	1	3	1	8

Dimension	1	2	6	10	20	30
Irr. complex chars.	6	15	220	0	0	0	241
Irr. rational chars.	2	2	0	1	1	44	50

Minimal presentations

Permutation degree:	$33$
Transitive degree:	$33$
Rank:	$2$
Inequivalent generating pairs:	$12$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	6	6	30
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid a^{6}=b^{11}=c^{11}=d^{11}=[b,c]=[b,d]=[c,d]=1, b^{a}=b^{10}c^{5}d^{7}, c^{a}=c^{9}d^{2}, d^{a}=c^{2}d^{3} \rangle$
Permutation group:	Degree $33$ $\langle(1,2,5,4,10,23)(3,6,14,11,20,27)(7,17,24,29,12,9)(8,21,31)(13,26,18,30,33,16) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 9 & 7 & 0 & 2 \\ 9 & 6 & 5 & 3 \\ 1 & 9 & 10 & 1 \\ 1 & 1 & 4 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 9 & 6 & 10 \\ 9 & 5 & 6 & 6 \\ 1 & 3 & 8 & 2 \\ 0 & 1 & 2 & 2 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 4 & 5 & 1 & 0 \\ 6 & 6 & 8 & 0 \\ 6 & 6 & 7 & 1 \end{array}\right), \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 9 & 10 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 2 & 0 & 2 & 10 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 1 & 7 \\ 6 & 8 & 9 & 1 \\ 8 & 9 & 5 & 1 \\ 0 & 8 & 5 & 9 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$C_{11}^3$ $\,\rtimes\,$ $C_6$	$(C_{11}\wr C_3)$ $\,\rtimes\,$ $C_2$	$(C_{11}^3:C_2)$ $\,\rtimes\,$ $C_3$	$(C_{11}^2:C_3)$ $\,\rtimes\,$ $D_{11}$	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:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{11}$
Commutator length:	$1$

Subgroups

There are 20864 subgroups in 192 conjugacy classes, 8 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_{11}^3:C_6$
Commutator:	$G' \simeq$ $C_{11}^3$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{11}^3:C_6$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}^3$	$G/\operatorname{Fit} \simeq$ $C_6$
Radical:	$R \simeq$ $C_{11}^3:C_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}^3$	$G/\operatorname{soc} \simeq$ $C_6$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
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

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 7986: $C_{11}^3:C_6$

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

Order 2662: $C_{11}^3:C_2$

Order 1331: $C_{11}^3$

Order 726: $C_{11}^2:C_6$

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

Order 242: $C_{11}:D_{11}$ x 45

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

Order 66: $C_3\times D_{11}$

Order 33: $C_{33}$

Order 22: $D_{11}$ x 45

Order 11: $C_{11}$ x 45

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 7986: $C_{11}^3:C_6$

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

Order 2662: $C_{11}^3:C_2$

Order 1331: $C_{11}^3$

Order 726: $C_{11}^2:C_6$

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

Order 242: $C_{11}:D_{11}$ x 3

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

Order 66: $C_3\times D_{11}$

Order 33: $C_{33}$

Order 22: $D_{11}$ x 3

Order 11: $C_{11}$ x 3

Order 6: $C_6$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 7986: $C_{11}^3:C_6$ ($C_1$)

Order 3993: $C_{11}\wr C_3$ ($C_2$)

Order 2662: $C_{11}^3:C_2$ ($C_3$)

Order 1331: $C_{11}^3$ ($C_6$)

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

Order 121: $C_{11}^2$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($C_{11}^2:C_6$)

Order 1: $C_1$ ($C_{11}^3:C_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 7986: $C_{11}^3:C_6$ ($C_1$)

Order 3993: $C_{11}\wr C_3$ ($C_2$)

Order 2662: $C_{11}^3:C_2$ ($C_3$)

Order 1331: $C_{11}^3$ ($C_6$)

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

Order 121: $C_{11}^2$ ($C_3\times D_{11}$)

Order 11: $C_{11}$ ($C_{11}^2:C_6$)

Order 1: $C_1$ ($C_{11}^3:C_6$)

Series

Derived series

$C_{11}^3:C_6$

$\rhd$

$C_{11}^3:C_6$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}^3:C_6$

$\rhd$

$C_{11}^3:C_6$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}\wr C_3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}^3:C_6$

$\rhd$

$C_{11}^3:C_6$

$\rhd$

$C_{11}^3$

$\rhd$

$C_{11}^3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

See the $241 \times 241$ 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 $50 \times 50$ rational character table.