Abstract group 1760.1091: $C_2\times D_4:F_{11}$

• Label: 1760.1091
• Order: \( 2^{5} \cdot 5 \cdot 11 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 11 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \cdot 5 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 5 \cdot 11 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \)
• Perm deg.: $21$
• Trans deg.: $176$
• Rank: $4$

Group information

Description:	$C_2\times D_4:F_{11}$
Order:	\(1760\)\(\medspace = 2^{5} \cdot 5 \cdot 11 \)
Exponent:	\(220\)\(\medspace = 2^{2} \cdot 5 \cdot 11 \)
Automorphism group:	$(C_2^3\times C_{22}).C_5.C_2^5$, of order \(28160\)\(\medspace = 2^{9} \cdot 5 \cdot 11 \)
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	10	11	20	22	44
Elements	1	55	136	44	660	10	704	110	40	1760
Conjugacy classes	1	9	10	4	36	1	40	7	2	110
Divisions	1	9	8	1	9	1	8	7	2	46
Autjugacy classes	1	4	3	4	16	1	12	3	1	45

Dimension	1	2	4	10	16	20
Irr. complex chars.	80	20	0	8	0	2	110
Irr. rational chars.	16	0	18	8	2	2	46

Minimal presentations

Permutation degree:	$21$
Transitive degree:	$176$
Rank:	$4$
Inequivalent generating quadruples:	$104569920$

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	$\langle a, b, c, d \mid a^{10}=b^{22}=c^{2}=d^{4}=[a,c]=[a,d]=[b,c]=[c,d]=1, b^{a}=b^{7}d^{2}, d^{b}=d^{3} \rangle$
Permutation group:	Degree $21$ $\langle(2,3)(4,6)(5,8)(7,11)(9,10)(12,13)(14,16)(15,17)(18,19)(20,21), (12,14) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 16 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 45 & 33 \\ 22 & 45 \end{array}\right), \left(\begin{array}{rr} 12 & 35 \\ 0 & 32 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 21 & 0 \\ 0 & 21 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/55\Z)$
Direct product:	$C_2$ $\, \times\, $ $(D_4:F_{11})$
Semidirect product:	$(D_4:D_{22})$ $\,\rtimes\,$ $C_5$	$(C_2\times D_4)$ $\,\rtimes\,$ $F_{11}$	$D_4$ $\,\rtimes\,$ $(C_2\times F_{11})$	$(C_{22}:Q_8)$ $\,\rtimes\,$ $C_{10}$	all 28
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_2\times F_{11})$ . $C_2^3$	$C_2^3$ . $(C_2\times F_{11})$	$C_4$ . $(C_2^2\times F_{11})$	$C_2$ . $(C_2^3\times F_{11})$	all 23

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

Homology

Abelianization:	$C_{2}^{3} \times C_{10} \simeq C_{2}^{4} \times C_{5}$
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 1906 subgroups in 328 conjugacy classes, 159 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $C_2^2\times F_{11}$
Commutator:	$G' \simeq$ $C_{22}$	$G/G' \simeq$ $C_2^3\times C_{10}$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^3\times F_{11}$
Fitting:	$\operatorname{Fit} \simeq$ $D_4\times C_{22}$	$G/\operatorname{Fit} \simeq$ $C_{10}$
Radical:	$R \simeq$ $C_2\times D_4:F_{11}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{22}$	$G/\operatorname{soc} \simeq$ $C_2^2\times C_{10}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4:C_2^2$
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

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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 1760: $C_2\times D_4:F_{11}$

Order 880: $D_4:F_{11}$ x 8, $C_2^3:F_{11}$ x 2, $C_2^3.F_{11}$ x 2, $C_2\times C_{44}:C_{10}$, $D_{22}:C_{20}$, $C_2\times C_{44}.C_{10}$

Order 440: $C_{22}:C_{20}$ x 11, $D_{22}:C_{10}$ x 8, $C_4\times F_{11}$ x 4, $C_{44}.C_{10}$ x 4, $C_{44}:C_{10}$ x 4, $C_2\times C_{22}:C_{10}$ x 2, $C_2^2\times F_{11}$, $C_{22}:C_{20}$

Order 352: $D_4:D_{22}$

Order 220: $C_{22}:C_{10}$ x 9, $C_{11}:C_{20}$ x 6, $C_2\times F_{11}$ x 4, $C_{11}:C_{20}$ x 2

Order 176: $D_4:D_{11}$ x 8, $C_{22}:D_4$ x 2, $C_2^2.D_{22}$ x 2, $D_4\times C_{22}$, $C_4\times D_{22}$, $C_{22}:Q_8$

Order 160: $C_{10}.C_2^4$

Order 110: $C_{11}:C_{10}$ x 7, $F_{11}$ x 2

Order 88: $C_{22}:C_4$ x 11, $C_{11}:D_4$ x 8, $D_4\times C_{11}$ x 4, $C_4\times D_{11}$ x 4, $C_{11}:Q_8$ x 4, $C_2^2\times C_{22}$ x 2, $C_2\times C_{44}$, $C_2\times D_{22}$

Order 80: $D_4:C_{10}$ x 8, $D_4\times C_{10}$ x 3, $C_2^2\times C_{20}$ x 3, $Q_8\times C_{10}$

Order 55: $C_{11}:C_5$

Order 44: $C_2\times C_{22}$ x 9, $C_{11}:C_4$ x 6, $D_{22}$ x 4, $C_{44}$ x 2

Order 40: $C_2\times C_{20}$ x 16, $C_5\times D_4$ x 12, $C_5\times Q_8$ x 4, $C_2^2\times C_{10}$ x 3

Order 32: $D_4:C_2^2$

Order 22: $C_{22}$ x 7, $D_{11}$ x 2

Order 20: $C_2\times C_{10}$ x 13, $C_{20}$ x 8

Order 16: $D_4:C_2$ x 8, $C_2\times D_4$ x 3, $C_2^2\times C_4$ x 3, $C_2\times Q_8$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 9

Order 8: $C_2\times C_4$ x 16, $D_4$ x 12, $Q_8$ x 4, $C_2^3$ x 3

Order 5: $C_5$

Order 4: $C_2^2$ x 13, $C_4$ x 8

Order 2: $C_2$ x 9

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1760: $C_2\times D_4:F_{11}$

Order 880: $C_2\times C_{44}:C_{10}$, $C_2^3:F_{11}$, $C_2^3.F_{11}$, $D_4:F_{11}$, $D_{22}:C_{20}$, $C_2\times C_{44}.C_{10}$

Order 440: $C_{22}:C_{20}$ x 3, $C_4\times F_{11}$, $C_{44}.C_{10}$, $C_2\times C_{22}:C_{10}$, $C_2^2\times F_{11}$, $C_{44}:C_{10}$, $C_{22}:C_{20}$, $D_{22}:C_{10}$

Order 352: $D_4:D_{22}$

Order 220: $C_{22}:C_{10}$ x 3, $C_2\times F_{11}$ x 2, $C_{11}:C_{20}$ x 2, $C_{11}:C_{20}$

Order 176: $D_4\times C_{22}$, $C_{22}:D_4$, $C_2^2.D_{22}$, $D_4:D_{11}$, $C_4\times D_{22}$, $C_{22}:Q_8$

Order 160: $C_{10}.C_2^4$

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

Order 88: $C_{22}:C_4$ x 3, $D_4\times C_{11}$, $C_2\times C_{44}$, $C_{11}:D_4$, $C_4\times D_{11}$, $C_{11}:Q_8$, $C_2^2\times C_{22}$, $C_2\times D_{22}$

Order 80: $D_4\times C_{10}$ x 2, $C_2^2\times C_{20}$ x 2, $D_4:C_{10}$, $Q_8\times C_{10}$

Order 55: $C_{11}:C_5$

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

Order 40: $C_2\times C_{20}$ x 5, $C_2^2\times C_{10}$ x 2, $C_5\times D_4$ x 2, $C_5\times Q_8$

Order 32: $D_4:C_2^2$

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

Order 20: $C_2\times C_{10}$ x 5, $C_{20}$ x 3

Order 16: $C_2\times D_4$ x 2, $C_2^2\times C_4$ x 2, $D_4:C_2$, $C_2\times Q_8$

Order 11: $C_{11}$

Order 10: $C_{10}$ x 4

Order 8: $C_2\times C_4$ x 5, $C_2^3$ x 2, $D_4$ x 2, $Q_8$

Order 5: $C_5$

Order 4: $C_2^2$ x 5, $C_4$ x 3

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1760: $C_2\times D_4:F_{11}$ ($C_1$)

Order 880: $D_4:F_{11}$ ($C_2$) x 8, $C_2^3:F_{11}$ ($C_2$) x 2, $C_2^3.F_{11}$ ($C_2$) x 2, $C_2\times C_{44}:C_{10}$ ($C_2$), $D_{22}:C_{20}$ ($C_2$), $C_2\times C_{44}.C_{10}$ ($C_2$)

Order 440: $C_{22}:C_{20}$ ($C_2^2$) x 11, $D_{22}:C_{10}$ ($C_2^2$) x 8, $C_4\times F_{11}$ ($C_2^2$) x 4, $C_{44}.C_{10}$ ($C_2^2$) x 4, $C_{44}:C_{10}$ ($C_2^2$) x 4, $C_2\times C_{22}:C_{10}$ ($C_2^2$) x 2, $C_2^2\times F_{11}$ ($C_2^2$), $C_{22}:C_{20}$ ($C_2^2$)

Order 352: $D_4:D_{22}$ ($C_5$)

Order 220: $C_{11}:C_{20}$ ($C_2^3$) x 6, $C_{22}:C_{10}$ ($C_2^3$) x 5, $C_2\times F_{11}$ ($C_2^3$) x 2, $C_{11}:C_{20}$ ($C_2^3$) x 2

Order 176: $D_4:D_{11}$ ($C_{10}$) x 8, $C_{22}:D_4$ ($C_{10}$) x 2, $C_2^2.D_{22}$ ($C_{10}$) x 2, $D_4\times C_{22}$ ($C_{10}$), $C_4\times D_{22}$ ($C_{10}$), $C_{22}:Q_8$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($D_4:C_2$) x 2, $C_{11}:C_{10}$ ($C_2^4$)

Order 88: $C_{22}:C_4$ ($C_2\times C_{10}$) x 11, $C_{11}:D_4$ ($C_2\times C_{10}$) x 8, $D_4\times C_{11}$ ($C_2\times C_{10}$) x 4, $C_4\times D_{11}$ ($C_2\times C_{10}$) x 4, $C_{11}:Q_8$ ($C_2\times C_{10}$) x 4, $C_2^2\times C_{22}$ ($C_2\times C_{10}$) x 2, $C_2\times C_{44}$ ($C_2\times C_{10}$), $C_2\times D_{22}$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($D_4:C_2^2$)

Order 44: $C_{11}:C_4$ ($C_2^2\times C_{10}$) x 6, $C_2\times C_{22}$ ($C_2^2\times C_{10}$) x 5, $D_{22}$ ($C_2^2\times C_{10}$) x 2, $C_{44}$ ($C_2^2\times C_{10}$) x 2

Order 22: $C_{22}$ ($D_4:C_{10}$) x 2, $C_{22}$ ($C_2^3\times C_{10}$)

Order 16: $C_2\times D_4$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{10}.C_2^4$)

Order 8: $D_4$ ($C_2\times F_{11}$) x 4, $C_2^3$ ($C_2\times F_{11}$) x 2, $C_2\times C_4$ ($C_2\times F_{11}$)

Order 4: $C_2^2$ ($C_2^2\times F_{11}$) x 5, $C_4$ ($C_2^2\times F_{11}$) x 2

Order 2: $C_2$ ($D_4:F_{11}$) x 2, $C_2$ ($C_2^3\times F_{11}$)

Order 1: $C_1$ ($C_2\times D_4:F_{11}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1760: $C_2\times D_4:F_{11}$ ($C_1$)

Order 880: $C_2\times C_{44}:C_{10}$ ($C_2$), $C_2^3:F_{11}$ ($C_2$), $C_2^3.F_{11}$ ($C_2$), $D_4:F_{11}$ ($C_2$), $D_{22}:C_{20}$ ($C_2$), $C_2\times C_{44}.C_{10}$ ($C_2$)

Order 440: $C_{22}:C_{20}$ ($C_2^2$) x 3, $C_4\times F_{11}$ ($C_2^2$), $C_{44}.C_{10}$ ($C_2^2$), $C_2\times C_{22}:C_{10}$ ($C_2^2$), $C_2^2\times F_{11}$ ($C_2^2$), $C_{44}:C_{10}$ ($C_2^2$), $C_{22}:C_{20}$ ($C_2^2$), $D_{22}:C_{10}$ ($C_2^2$)

Order 352: $D_4:D_{22}$ ($C_5$)

Order 220: $C_{22}:C_{10}$ ($C_2^3$) x 2, $C_{11}:C_{20}$ ($C_2^3$) x 2, $C_2\times F_{11}$ ($C_2^3$), $C_{11}:C_{20}$ ($C_2^3$)

Order 176: $D_4\times C_{22}$ ($C_{10}$), $C_{22}:D_4$ ($C_{10}$), $C_2^2.D_{22}$ ($C_{10}$), $D_4:D_{11}$ ($C_{10}$), $C_4\times D_{22}$ ($C_{10}$), $C_{22}:Q_8$ ($C_{10}$)

Order 110: $C_{11}:C_{10}$ ($C_2^4$), $C_{11}:C_{10}$ ($D_4:C_2$)

Order 88: $C_{22}:C_4$ ($C_2\times C_{10}$) x 3, $D_4\times C_{11}$ ($C_2\times C_{10}$), $C_2\times C_{44}$ ($C_2\times C_{10}$), $C_{11}:D_4$ ($C_2\times C_{10}$), $C_4\times D_{11}$ ($C_2\times C_{10}$), $C_{11}:Q_8$ ($C_2\times C_{10}$), $C_2^2\times C_{22}$ ($C_2\times C_{10}$), $C_2\times D_{22}$ ($C_2\times C_{10}$)

Order 55: $C_{11}:C_5$ ($D_4:C_2^2$)

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

Order 22: $C_{22}$ ($C_2^3\times C_{10}$), $C_{22}$ ($D_4:C_{10}$)

Order 16: $C_2\times D_4$ ($F_{11}$)

Order 11: $C_{11}$ ($C_{10}.C_2^4$)

Order 8: $C_2^3$ ($C_2\times F_{11}$), $D_4$ ($C_2\times F_{11}$), $C_2\times C_4$ ($C_2\times F_{11}$)

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

Order 2: $C_2$ ($C_2^3\times F_{11}$), $C_2$ ($D_4:F_{11}$)

Order 1: $C_1$ ($C_2\times D_4:F_{11}$)

Series

Derived series

$C_2\times D_4:F_{11}$

$\rhd$

$C_{22}$

$\rhd$

$C_1$

Chief series

$C_2\times D_4:F_{11}$

$\rhd$

$C_2\times C_{44}:C_{10}$

$\rhd$

$D_4\times C_{22}$

$\rhd$

$C_2\times C_{44}$

$\rhd$

$C_2\times C_4$

$\rhd$

$C_4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2\times D_4:F_{11}$

$\rhd$

$C_{22}$

$\rhd$

$C_{11}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times D_4$

Character theory

Complex character table

See the $110 \times 110$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $46 \times 46$ rational character table.