Abstract group 1760.355: $C_{11}:(C_{10}\times Q_{16})$

• Label: 1760.355
• Order: \( 2^{5} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{3} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $29$
• Trans deg.: $352$
• Rank: $3$

Group information

Description:	$C_{11}:(C_{10}\times Q_{16})$
Order:	\(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \)
Exponent:	\(440\)\(\medspace = 2^{3} \cdot 5 \cdot 11 \)
Automorphism group:	$C_{11}.(C_{10}\times D_4).C_2^4$, of order \(14080\)\(\medspace = 2^{8} \cdot 5 \cdot 11 \)
Composition factors:	$C_2$ x 5, $C_5$, $C_{11}$
Derived length:	$2$

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

Group statistics

Order	1	2	4	5	8	10	11	20	22	40	44
Elements	1	3	100	44	88	132	10	880	30	352	120	1760
Conjugacy classes	1	3	6	4	4	12	1	24	3	16	6	80
Divisions	1	3	6	1	2	3	1	6	3	2	4	32
Autjugacy classes	1	2	4	4	1	8	1	16	2	4	3	46

Dimension	1	2	4	8	10	16	20
Irr. complex chars.	40	30	0	0	8	0	2	80
Irr. rational chars.	8	2	10	2	4	2	4	32

Minimal presentations

Permutation degree:	$29$
Transitive degree:	$352$
Rank:	$3$
Inequivalent generating triples:	$124992$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	12	14	18

Constructions

Presentation:	$\langle a, b, c \mid b^{40}=c^{22}=[a,c]=1, a^{2}=b^{20}, b^{a}=b^{31}, c^{b}=c^{17} \rangle$
Permutation group:	Degree $29$ $\langle(2,3)(4,5)(6,7)(8,10)(9,11)(12,13,16,19)(14,20,22,27)(15,23,25,18)(17,24,26,21) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 43 & 64 \\ 33 & 34 \end{array}\right), \left(\begin{array}{rr} 15 & 0 \\ 0 & 64 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 45 & 55 \\ 66 & 67 \end{array}\right), \left(\begin{array}{rr} 23 & 33 \\ 55 & 12 \end{array}\right), \left(\begin{array}{rr} 43 & 0 \\ 0 & 43 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/77\Z)$
Direct product:	$C_2$ $\, \times\, $ $(Q_8.F_{11})$
Semidirect product:	$(C_{22}:Q_{16})$ $\,\rtimes\,$ $C_5$	$C_{22}$ $\,\rtimes\,$ $(C_5\times Q_{16})$ (2)	$(C_{11}:Q_{16})$ $\,\rtimes\,$ $C_{10}$ (4)	$(C_{11}:C_{10})$ $\,\rtimes\,$ $Q_{16}$ (2)	all 6
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times Q_8)$ . $F_{11}$	$Q_8$ . $(C_2\times F_{11})$ (2)	$C_{44}$ . $(C_5\times D_4)$	$(Q_8\times C_{22})$ . $C_{10}$	all 29

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 722 subgroups in 120 conjugacy classes, 55 normal (27 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{22}:C_{10}$
Commutator:	$G' \simeq$ $C_{44}$	$G/G' \simeq$ $C_2^2\times C_{10}$
Frattini:	$\Phi \simeq$ $C_4$	$G/\Phi \simeq$ $C_2^2\times F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_{10}$
Radical:	$R \simeq$ $C_{11}:(C_{10}\times Q_{16})$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_5\times D_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times Q_{16}$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
11-Sylow subgroup:	$P_{ 11 } \simeq$ $C_{11}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1760: $C_{11}:(C_{10}\times Q_{16})$

Order 880: $Q_8.F_{11}$ x 4, $Q_8\times C_{11}:C_{10}$, $C_{22}:C_{40}$, $C_2\times C_{44}.C_{10}$

Order 440: $C_{44}.C_{10}$ x 3, $C_{44}.C_{10}$ x 3, $C_{22}:C_{20}$ x 2, $C_{11}:C_{40}$ x 2, $C_{22}:C_{20}$

Order 352: $C_{22}:Q_{16}$

Order 220: $C_{11}:C_{20}$ x 4, $C_{11}:C_{20}$ x 2, $C_{22}:C_{10}$

Order 176: $C_{11}:Q_{16}$ x 4, $C_{22}:C_8$, $Q_8\times C_{22}$, $C_{22}:Q_8$

Order 160: $C_{10}\times Q_{16}$

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

Order 88: $C_{11}:Q_8$ x 3, $Q_8\times C_{11}$ x 3, $C_2\times C_{44}$ x 2, $C_{11}:C_8$ x 2, $C_{22}:C_4$

Order 80: $C_5\times Q_{16}$ x 4, $Q_8\times C_{10}$ x 2, $C_2\times C_{40}$

Order 55: $C_{11}:C_5$

Order 44: $C_{44}$ x 4, $C_{11}:C_4$ x 2, $C_2\times C_{22}$

Order 40: $C_5\times Q_8$ x 6, $C_2\times C_{20}$ x 3, $C_{40}$ x 2

Order 32: $C_2\times Q_{16}$

Order 22: $C_{22}$ x 3

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

Order 16: $Q_{16}$ x 4, $C_2\times Q_8$ x 2, $C_2\times C_8$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 3

Order 8: $Q_8$ x 6, $C_2\times C_4$ x 3, $C_8$ x 2

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 1760: $C_{11}:(C_{10}\times Q_{16})$

Order 880: $Q_8.F_{11}$, $Q_8\times C_{11}:C_{10}$, $C_{22}:C_{40}$, $C_2\times C_{44}.C_{10}$

Order 440: $C_{44}.C_{10}$ x 2, $C_{44}.C_{10}$ x 2, $C_{22}:C_{20}$ x 2, $C_{22}:C_{20}$, $C_{11}:C_{40}$

Order 352: $C_{22}:Q_{16}$

Order 220: $C_{11}:C_{20}$ x 3, $C_{22}:C_{10}$, $C_{11}:C_{20}$

Order 176: $C_{22}:C_8$, $Q_8\times C_{22}$, $C_{22}:Q_8$, $C_{11}:Q_{16}$

Order 160: $C_{10}\times Q_{16}$

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

Order 88: $C_2\times C_{44}$ x 2, $C_{11}:Q_8$ x 2, $Q_8\times C_{11}$ x 2, $C_{22}:C_4$, $C_{11}:C_8$

Order 80: $Q_8\times C_{10}$ x 2, $C_5\times Q_{16}$, $C_2\times C_{40}$

Order 55: $C_{11}:C_5$

Order 44: $C_{44}$ x 3, $C_2\times C_{22}$, $C_{11}:C_4$

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

Order 32: $C_2\times Q_{16}$

Order 22: $C_{22}$ x 2

Order 20: $C_{20}$ x 4, $C_2\times C_{10}$

Order 16: $C_2\times Q_8$ x 2, $Q_{16}$, $C_2\times C_8$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 2

Order 8: $Q_8$ x 4, $C_2\times C_4$ x 3, $C_8$

Order 5: $C_5$

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

Order 2: $C_2$ x 2

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1760: $C_{11}:(C_{10}\times Q_{16})$ ($C_1$)

Order 880: $Q_8.F_{11}$ ($C_2$) x 4, $Q_8\times C_{11}:C_{10}$ ($C_2$), $C_{22}:C_{40}$ ($C_2$), $C_2\times C_{44}.C_{10}$ ($C_2$)

Order 440: $C_{44}.C_{10}$ ($C_2^2$) x 2, $C_{44}.C_{10}$ ($C_2^2$) x 2, $C_{11}:C_{40}$ ($C_2^2$) x 2, $C_{22}:C_{20}$ ($C_2^2$)

Order 352: $C_{22}:Q_{16}$ ($C_5$)

Order 220: $C_{22}:C_{10}$ ($D_4$), $C_{11}:C_{20}$ ($C_2^3$), $C_{11}:C_{20}$ ($D_4$)

Order 176: $C_{11}:Q_{16}$ ($C_{10}$) x 4, $C_{22}:C_8$ ($C_{10}$), $Q_8\times C_{22}$ ($C_{10}$), $C_{22}:Q_8$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($Q_{16}$) x 2, $C_{11}:C_{10}$ ($C_2\times D_4$)

Order 88: $C_{11}:Q_8$ ($C_2\times C_{10}$) x 2, $Q_8\times C_{11}$ ($C_2\times C_{10}$) x 2, $C_{11}:C_8$ ($C_2\times C_{10}$) x 2, $C_2\times C_{44}$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($C_2\times Q_{16}$)

Order 44: $C_2\times C_{22}$ ($C_5\times D_4$), $C_{44}$ ($C_2^2\times C_{10}$), $C_{44}$ ($C_5\times D_4$)

Order 22: $C_{22}$ ($C_5\times Q_{16}$) x 2, $C_{22}$ ($D_4\times C_{10}$)

Order 16: $C_2\times Q_8$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{10}\times Q_{16}$)

Order 8: $Q_8$ ($C_2\times F_{11}$) x 2, $C_2\times C_4$ ($C_2\times F_{11}$)

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

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

Order 1: $C_1$ ($C_{11}:(C_{10}\times Q_{16})$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1760: $C_{11}:(C_{10}\times Q_{16})$ ($C_1$)

Order 880: $Q_8.F_{11}$ ($C_2$), $Q_8\times C_{11}:C_{10}$ ($C_2$), $C_{22}:C_{40}$ ($C_2$), $C_2\times C_{44}.C_{10}$ ($C_2$)

Order 440: $C_{44}.C_{10}$ ($C_2^2$), $C_{44}.C_{10}$ ($C_2^2$), $C_{22}:C_{20}$ ($C_2^2$), $C_{11}:C_{40}$ ($C_2^2$)

Order 352: $C_{22}:Q_{16}$ ($C_5$)

Order 220: $C_{22}:C_{10}$ ($D_4$), $C_{11}:C_{20}$ ($C_2^3$), $C_{11}:C_{20}$ ($D_4$)

Order 176: $C_{22}:C_8$ ($C_{10}$), $Q_8\times C_{22}$ ($C_{10}$), $C_{22}:Q_8$ ($C_{10}$), $C_{11}:Q_{16}$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($Q_{16}$), $C_{11}:C_{10}$ ($C_2\times D_4$)

Order 88: $C_2\times C_{44}$ ($C_2\times C_{10}$), $C_{11}:Q_8$ ($C_2\times C_{10}$), $Q_8\times C_{11}$ ($C_2\times C_{10}$), $C_{11}:C_8$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($C_2\times Q_{16}$)

Order 44: $C_2\times C_{22}$ ($C_5\times D_4$), $C_{44}$ ($C_2^2\times C_{10}$), $C_{44}$ ($C_5\times D_4$)

Order 22: $C_{22}$ ($D_4\times C_{10}$), $C_{22}$ ($C_5\times Q_{16}$)

Order 16: $C_2\times Q_8$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{10}\times Q_{16}$)

Order 8: $Q_8$ ($C_2\times F_{11}$), $C_2\times C_4$ ($C_2\times F_{11}$)

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

Order 2: $C_2$ ($Q_8.F_{11}$), $C_2$ ($C_2^3:F_{11}$)

Order 1: $C_1$ ($C_{11}:(C_{10}\times Q_{16})$)

Series

Derived series

$C_{11}:(C_{10}\times Q_{16})$

$\rhd$

$C_{44}$

$\rhd$

$C_1$

Chief series

$C_{11}:(C_{10}\times Q_{16})$

$\rhd$

$Q_8\times C_{11}:C_{10}$

$\rhd$

$C_{22}:C_{20}$

$\rhd$

$C_{22}:C_{10}$

$\rhd$

$C_{11}:C_{10}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

$\rhd$

$C_1$

Lower central series

$C_{11}:(C_{10}\times Q_{16})$

$\rhd$

$C_{44}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

$\lhd$

$C_2\times Q_8$

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 $32 \times 32$ rational character table.