Abstract group 9680.p: $C_{11}:(Q_8\times F_{11})$

• Label: 9680.p
• Order: \( 2^{4} \cdot 5 \cdot 11^{2} \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 5^{2} \cdot 11^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 5 \)
• Perm deg.: $30$
• Trans deg.: $88$
• Rank: $3$

Group information

Description:	$C_{11}:(Q_8\times F_{11})$
Order:	\(9680\)\(\medspace = 2^{4} \cdot 5 \cdot 11^{2} \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}^2.C_{10}^2.C_2^5$, of order \(387200\)\(\medspace = 2^{7} \cdot 5^{2} \cdot 11^{2} \)
Composition factors:	$C_2$ x 4, $C_5$, $C_{11}$ x 2
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	10	11	20	22	44
Elements	1	243	332	484	1452	120	5808	120	1120	9680
Conjugacy classes	1	3	6	4	12	7	24	7	16	80
Divisions	1	3	6	1	3	7	6	7	11	45
Autjugacy classes	1	2	3	4	8	2	12	2	3	37

Dimension	1	2	4	8	10	20	40
Irr. complex chars.	40	10	0	0	8	22	0	80
Irr. rational chars.	8	2	8	2	8	12	5	45

Minimal presentations

Permutation degree:	$30$
Transitive degree:	$88$
Rank:	$3$
Inequivalent generating triples:	$749952$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	20	40	40
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c \mid a^{10}=c^{44}=1, b^{22}=c^{22}, b^{a}=b^{13}c^{8}, c^{a}=c^{13}, c^{b}=c^{43} \rangle$
Permutation group:	Degree $30$ $\langle(1,2,6,9,11,8,3,4,7,10)(12,13,14,16)(15,19,18,17)(20,21,26,25,30)(22,23,28,24,29) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rrrr} 0 & 9 & 2 & 2 \\ 3 & 8 & 0 & 4 \\ 0 & 1 & 10 & 0 \\ 4 & 5 & 6 & 4 \end{array}\right), \left(\begin{array}{rrrr} 1 & 8 & 3 & 10 \\ 9 & 4 & 0 & 3 \\ 6 & 9 & 5 & 3 \\ 1 & 6 & 2 & 8 \end{array}\right), \left(\begin{array}{rrrr} 0 & 9 & 6 & 10 \\ 10 & 6 & 9 & 6 \\ 8 & 10 & 7 & 2 \\ 7 & 8 & 1 & 2 \end{array}\right), \left(\begin{array}{rrrr} 8 & 7 & 4 & 6 \\ 5 & 5 & 1 & 4 \\ 0 & 4 & 4 & 4 \\ 6 & 0 & 6 & 1 \end{array}\right), \left(\begin{array}{rrrr} 6 & 7 & 4 & 1 \\ 2 & 1 & 2 & 6 \\ 6 & 1 & 3 & 0 \\ 5 & 1 & 8 & 1 \end{array}\right), \left(\begin{array}{rrrr} 4 & 10 & 1 & 7 \\ 1 & 3 & 5 & 1 \\ 6 & 7 & 10 & 1 \\ 9 & 6 & 10 & 9 \end{array}\right), \left(\begin{array}{rrrr} 10 & 0 & 0 & 0 \\ 0 & 10 & 0 & 0 \\ 0 & 0 & 10 & 0 \\ 0 & 0 & 0 & 10 \end{array}\right) \right\rangle \subseteq \GL_{4}(\F_{11})$
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_{11}:F_{11})$ $\,\rtimes\,$ $Q_8$	$(C_{11}:Q_8)$ $\,\rtimes\,$ $F_{11}$	$C_{11}$ $\,\rtimes\,$ $(Q_8\times F_{11})$	$(C_{44}.D_{22})$ $\,\rtimes\,$ $C_5$	all 11
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_4$ . $(D_{11}:F_{11})$	$C_2$ . $(D_{22}:F_{11})$	$(C_{44}:F_{11})$ . $C_2$	$C_{44}$ . $(C_2\times F_{11})$	all 19
Aut. group:	$\Aut(C_4^3.C_2\wr S_3)$

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

Homology

Abelianization:	$C_{2}^{2} \times C_{10} \simeq C_{2}^{3} \times C_{5}$
Schur multiplier:	$C_{2}^{2}$
Commutator length:	$1$

Subgroups

There are 5792 subgroups in 232 conjugacy classes, 53 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{22}:F_{11}$
Commutator:	$G' \simeq$ $C_{11}\times C_{22}$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{22}:F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{11}\times C_{44}$	$G/\operatorname{Fit} \simeq$ $C_2\times C_{10}$
Radical:	$R \simeq$ $C_{11}:(Q_8\times F_{11})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{11}\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_8$
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

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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Subgroup information

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Classes of subgroups up to conjugation

Order 9680: $C_{11}:(Q_8\times F_{11})$

Order 4840: $C_{44}.F_{11}$ x 2, $C_{11}^2:(C_5\times Q_8)$ x 2, $C_{11}:(C_4\times F_{11})$ x 2, $C_{44}:F_{11}$

Order 2420: $C_{11}^2:C_{20}$ x 4, $C_{11}^2:C_{20}$, $C_{22}:F_{11}$, $C_{11}^2:C_{20}$

Order 1936: $C_{44}.D_{22}$

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

Order 968: $C_{11}^2:Q_8$ x 2, $C_{11}^2:Q_8$ x 2, $C_{22}.D_{22}$ x 2, $C_{44}:D_{11}$

Order 880: $Q_8\times F_{11}$ x 2

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

Order 484: $C_{11}:C_{44}$ x 4, $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:D_{22}$

Order 440: $C_4\times F_{11}$ x 11, $C_{44}.C_{10}$ x 6, $C_{44}.C_{10}$ x 2

Order 242: $C_{11}:D_{11}$ x 2, $C_{11}\times C_{22}$

Order 220: $C_{11}:C_{20}$ x 11, $C_{11}:C_{20}$ x 11, $C_2\times F_{11}$ x 7

Order 176: $Q_8\times D_{11}$ x 2

Order 121: $C_{11}^2$

Order 110: $F_{11}$ x 14, $C_{11}:C_{10}$ x 7

Order 88: $C_4\times D_{11}$ x 11, $C_{11}:Q_8$ x 6, $Q_8\times C_{11}$ x 2

Order 80: $Q_8\times C_{10}$

Order 55: $C_{11}:C_5$ x 7

Order 44: $C_{44}$ x 11, $C_{11}:C_4$ x 11, $D_{22}$ x 7

Order 40: $C_5\times Q_8$ x 4, $C_2\times C_{20}$ x 3

Order 22: $D_{11}$ x 14, $C_{22}$ x 7

Order 20: $C_{20}$ x 6, $C_2\times C_{10}$

Order 16: $C_2\times Q_8$

Order 11: $C_{11}$ x 7

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3

Order 5: $C_5$

Order 4: $C_4$ x 6, $C_2^2$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 9680: $C_{11}:(Q_8\times F_{11})$

Order 4840: $C_{44}:F_{11}$, $C_{44}.F_{11}$, $C_{11}^2:(C_5\times Q_8)$, $C_{11}:(C_4\times F_{11})$

Order 2420: $C_{11}^2:C_{20}$, $C_{22}:F_{11}$, $C_{11}^2:C_{20}$, $C_{11}^2:C_{20}$

Order 1936: $C_{44}.D_{22}$

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

Order 968: $C_{44}:D_{11}$, $C_{11}^2:Q_8$, $C_{11}^2:Q_8$, $C_{22}.D_{22}$

Order 880: $Q_8\times F_{11}$

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

Order 484: $C_{11}\times C_{44}$, $C_{11}^2:C_4$, $C_{11}:C_{44}$, $C_{11}:D_{22}$

Order 440: $C_4\times F_{11}$ x 3, $C_{44}.C_{10}$ x 2, $C_{44}.C_{10}$

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

Order 220: $C_{11}:C_{20}$ x 3, $C_{11}:C_{20}$ x 3, $C_2\times F_{11}$ x 2

Order 176: $Q_8\times D_{11}$

Order 121: $C_{11}^2$

Order 110: $C_{11}:C_{10}$ x 2, $F_{11}$ x 2

Order 88: $C_4\times D_{11}$ x 3, $C_{11}:Q_8$ x 2, $Q_8\times C_{11}$

Order 80: $Q_8\times C_{10}$

Order 55: $C_{11}:C_5$ x 2

Order 44: $C_{44}$ x 3, $C_{11}:C_4$ x 3, $D_{22}$ x 2

Order 40: $C_2\times C_{20}$ x 2, $C_5\times Q_8$ x 2

Order 22: $C_{22}$ x 2, $D_{11}$ x 2

Order 20: $C_{20}$ x 3, $C_2\times C_{10}$

Order 16: $C_2\times Q_8$

Order 11: $C_{11}$ x 2

Order 10: $C_{10}$ x 2

Order 8: $Q_8$ x 2, $C_2\times C_4$ x 2

Order 5: $C_5$

Order 4: $C_4$ x 3, $C_2^2$

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 9680: $C_{11}:(Q_8\times F_{11})$ ($C_1$)

Order 4840: $C_{44}.F_{11}$ ($C_2$) x 2, $C_{11}^2:(C_5\times Q_8)$ ($C_2$) x 2, $C_{11}:(C_4\times F_{11})$ ($C_2$) x 2, $C_{44}:F_{11}$ ($C_2$)

Order 2420: $C_{11}^2:C_{20}$ ($C_2^2$) x 4, $C_{11}^2:C_{20}$ ($C_2^2$), $C_{22}:F_{11}$ ($C_2^2$), $C_{11}^2:C_{20}$ ($C_2^2$)

Order 1936: $C_{44}.D_{22}$ ($C_5$)

Order 1210: $C_{11}:F_{11}$ ($Q_8$) x 2, $C_{11}^2:C_{10}$ ($C_2^3$)

Order 968: $C_{11}^2:Q_8$ ($C_{10}$) x 2, $C_{11}^2:Q_8$ ($C_{10}$) x 2, $C_{22}.D_{22}$ ($C_{10}$) x 2, $C_{44}:D_{11}$ ($C_{10}$)

Order 605: $C_{11}^2:C_5$ ($C_2\times Q_8$)

Order 484: $C_{11}:C_{44}$ ($C_2\times C_{10}$) x 4, $C_{11}\times C_{44}$ ($C_2\times C_{10}$), $C_{11}^2:C_4$ ($C_2\times C_{10}$), $C_{11}:D_{22}$ ($C_2\times C_{10}$)

Order 242: $C_{11}:D_{11}$ ($C_5\times Q_8$) x 2, $C_{11}\times C_{22}$ ($C_2^2\times C_{10}$)

Order 121: $C_{11}^2$ ($Q_8\times C_{10}$)

Order 88: $C_{11}:Q_8$ ($F_{11}$) x 2

Order 44: $C_{11}:C_4$ ($C_2\times F_{11}$) x 4, $C_{44}$ ($C_2\times F_{11}$) x 2

Order 22: $C_{22}$ ($C_2^2\times F_{11}$) x 2

Order 11: $C_{11}$ ($Q_8\times F_{11}$) x 2

Order 4: $C_4$ ($D_{11}:F_{11}$)

Order 2: $C_2$ ($D_{22}:F_{11}$)

Order 1: $C_1$ ($C_{11}:(Q_8\times F_{11})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 9680: $C_{11}:(Q_8\times F_{11})$ ($C_1$)

Order 4840: $C_{44}:F_{11}$ ($C_2$), $C_{44}.F_{11}$ ($C_2$), $C_{11}^2:(C_5\times Q_8)$ ($C_2$), $C_{11}:(C_4\times F_{11})$ ($C_2$)

Order 2420: $C_{11}^2:C_{20}$ ($C_2^2$), $C_{22}:F_{11}$ ($C_2^2$), $C_{11}^2:C_{20}$ ($C_2^2$), $C_{11}^2:C_{20}$ ($C_2^2$)

Order 1936: $C_{44}.D_{22}$ ($C_5$)

Order 1210: $C_{11}:F_{11}$ ($Q_8$), $C_{11}^2:C_{10}$ ($C_2^3$)

Order 968: $C_{44}:D_{11}$ ($C_{10}$), $C_{11}^2:Q_8$ ($C_{10}$), $C_{11}^2:Q_8$ ($C_{10}$), $C_{22}.D_{22}$ ($C_{10}$)

Order 605: $C_{11}^2:C_5$ ($C_2\times Q_8$)

Order 484: $C_{11}\times C_{44}$ ($C_2\times C_{10}$), $C_{11}^2:C_4$ ($C_2\times C_{10}$), $C_{11}:C_{44}$ ($C_2\times C_{10}$), $C_{11}:D_{22}$ ($C_2\times C_{10}$)

Order 242: $C_{11}\times C_{22}$ ($C_2^2\times C_{10}$), $C_{11}:D_{11}$ ($C_5\times Q_8$)

Order 121: $C_{11}^2$ ($Q_8\times C_{10}$)

Order 88: $C_{11}:Q_8$ ($F_{11}$)

Order 44: $C_{44}$ ($C_2\times F_{11}$), $C_{11}:C_4$ ($C_2\times F_{11}$)

Order 22: $C_{22}$ ($C_2^2\times F_{11}$)

Order 11: $C_{11}$ ($Q_8\times F_{11}$)

Order 4: $C_4$ ($D_{11}:F_{11}$)

Order 2: $C_2$ ($D_{22}:F_{11}$)

Order 1: $C_1$ ($C_{11}:(Q_8\times F_{11})$)

Series

Derived series

$C_{11}:(Q_8\times F_{11})$

$\rhd$

$C_{11}:(Q_8\times F_{11})$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$C_{11}:(Q_8\times F_{11})$

$\rhd$

$C_{11}:(Q_8\times F_{11})$

$\rhd$

$C_{44}.F_{11}$

$\rhd$

$C_{44}.F_{11}$

$\rhd$

$C_{11}^2:Q_8$

$\rhd$

$C_{11}^2:Q_8$

$\rhd$

$C_{11}\times C_{44}$

$\rhd$

$C_{11}\times C_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_{11}:(Q_8\times F_{11})$

$\rhd$

$C_{11}:(Q_8\times F_{11})$

$\rhd$

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

$\rhd$

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

$\rhd$

$C_{11}^2$

$\rhd$

$C_{11}^2$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

$\lhd$

$C_4$

$\lhd$

$C_4$

Supergroups

This group is a maximal subgroup of 4 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 $80 \times 80$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.