Abstract group 560.131: $C_{14}:D_{20}$

• Label: 560.131
• Order: \( 2^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $280$
• Rank: $3$

Group information

Description:	$C_{14}:D_{20}$
Order:	\(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \)
Exponent:	\(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$(C_2\times C_{70}).C_6.C_2^5$, of order \(26880\)\(\medspace = 2^{8} \cdot 3 \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_5$, $C_7$
Derived length:	$2$

This group is nonabelian, supersolvable (hence solvable and monomial), hyperelementary for $p = 2$, and metabelian.

Group statistics

Order	1	2	4	5	7	10	14	20	35	70
Elements	1	163	28	4	6	12	138	112	24	72	560
Conjugacy classes	1	7	2	2	3	6	21	8	6	18	74
Divisions	1	7	2	1	1	3	5	2	1	3	26
Autjugacy classes	1	4	1	1	1	2	3	1	1	2	17

Dimension	1	2	4	6	8	12	24
Irr. complex chars.	8	42	24	0	0	0	0	74
Irr. rational chars.	8	2	4	4	2	2	4	26

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$280$
Rank:	$3$
Inequivalent generating triples:	$2016$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	4	4	12

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{20}=c^{14}=[a,c]=1, b^{a}=b^{19}, c^{b}=c^{13} \rangle$
Permutation group:	Degree $18$ $\langle(6,7,8,9)(11,12)(13,14)(15,16)(17,18), (2,3)(4,5)(7,9), (17,18), (6,8)(7,9), (1,2,4,5,3), (10,11,13,15,16,14,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 99 & 14 \\ 35 & 6 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$C_2$ $\, \times\, $ $(C_7:D_{20})$
Semidirect product:	$C_{70}$ $\,\rtimes\,$ $D_4$ (2)	$D_{10}$ $\,\rtimes\,$ $D_{14}$ (2)	$C_{14}$ $\,\rtimes\,$ $D_{20}$ (2)	$D_{70}$ $\,\rtimes\,$ $C_2^2$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_{70}$ . $C_2^3$	$C_2$ . $(D_5\times D_{14})$	$C_2^2$ . $(D_5\times D_7)$	$(C_2\times C_{14})$ . $D_{10}$	all 8

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{3}$
Commutator length:	$1$

Subgroups

There are 968 subgroups in 108 conjugacy classes, 40 normal (18 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_5\times D_7$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_5\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{70}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{14}:D_{20}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{70}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\times D_4$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

There are too many subgroups to show. See a full page version of the diagram.

Classes of subgroups up to automorphism

There are too many subgroups to show. See a full page version of the diagram.

Normal subgroups

Normal subgroups up to automorphism

For the default diagram, subgroups are sorted vertically by the number of prime divisors (counted with multiplicity) in their orders.
To see subgroups sorted vertically by order instead, check this box.

Sorry, your browser does not support the subgroup diagram.

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

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

See a full page version of the diagram

Classes of subgroups up to conjugation

Order 560: $C_{14}:D_{20}$

Order 280: $C_7:D_{20}$ x 4, $C_2\times D_{70}$, $C_{14}\times D_{10}$, $C_{14}:C_{20}$

Order 140: $C_7\times D_{10}$ x 4, $D_{70}$ x 4, $C_7:C_{20}$ x 2, $C_2\times C_{70}$

Order 112: $C_{14}:D_4$

Order 80: $C_2\times D_{20}$

Order 70: $C_{70}$ x 3, $D_{35}$ x 2, $C_7\times D_5$ x 2

Order 56: $C_7:D_4$ x 4, $C_{14}:C_4$, $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 40: $D_{20}$ x 4, $C_2\times D_{10}$ x 2, $C_2\times C_{20}$

Order 35: $C_{35}$

Order 28: $C_2\times C_{14}$ x 5, $D_{14}$ x 4, $C_7:C_4$ x 2

Order 20: $D_{10}$ x 8, $C_{20}$ x 2, $C_2\times C_{10}$

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 5, $D_7$ x 2

Order 10: $D_5$ x 4, $C_{10}$ x 3

Order 8: $D_4$ x 4, $C_2^3$ x 2, $C_2\times C_4$

Order 7: $C_7$

Order 5: $C_5$

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 560: $C_{14}:D_{20}$

Order 280: $C_2\times D_{70}$, $C_{14}\times D_{10}$, $C_{14}:C_{20}$, $C_7:D_{20}$

Order 140: $C_7\times D_{10}$ x 2, $D_{70}$ x 2, $C_7:C_{20}$, $C_2\times C_{70}$

Order 112: $C_{14}:D_4$

Order 80: $C_2\times D_{20}$

Order 70: $C_{70}$ x 2, $D_{35}$, $C_7\times D_5$

Order 56: $C_7:D_4$, $C_{14}:C_4$, $C_2^2\times C_{14}$, $C_2\times D_{14}$

Order 40: $C_2\times D_{10}$ x 2, $C_2\times C_{20}$, $D_{20}$

Order 35: $C_{35}$

Order 28: $C_2\times C_{14}$ x 3, $D_{14}$ x 2, $C_7:C_4$

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

Order 16: $C_2\times D_4$

Order 14: $C_{14}$ x 3, $D_7$

Order 10: $C_{10}$ x 2, $D_5$ x 2

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

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 560: $C_{14}:D_{20}$ ($C_1$)

Order 280: $C_7:D_{20}$ ($C_2$) x 4, $C_2\times D_{70}$ ($C_2$), $C_{14}\times D_{10}$ ($C_2$), $C_{14}:C_{20}$ ($C_2$)

Order 140: $C_7\times D_{10}$ ($C_2^2$) x 2, $C_7:C_{20}$ ($C_2^2$) x 2, $D_{70}$ ($C_2^2$) x 2, $C_2\times C_{70}$ ($C_2^2$)

Order 70: $C_{70}$ ($D_4$) x 2, $C_{70}$ ($C_2^3$)

Order 56: $C_{14}:C_4$ ($D_5$)

Order 40: $C_2\times D_{10}$ ($D_7$)

Order 35: $C_{35}$ ($C_2\times D_4$)

Order 28: $C_7:C_4$ ($D_{10}$) x 2, $C_2\times C_{14}$ ($D_{10}$)

Order 20: $D_{10}$ ($D_{14}$) x 2, $C_2\times C_{10}$ ($D_{14}$)

Order 14: $C_{14}$ ($D_{20}$) x 2, $C_{14}$ ($C_2\times D_{10}$)

Order 10: $C_{10}$ ($C_7:D_4$) x 2, $C_{10}$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_2\times D_{20}$)

Order 5: $C_5$ ($C_{14}:D_4$)

Order 4: $C_2^2$ ($D_5\times D_7$)

Order 2: $C_2$ ($C_7:D_{20}$) x 2, $C_2$ ($D_5\times D_{14}$)

Order 1: $C_1$ ($C_{14}:D_{20}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 560: $C_{14}:D_{20}$ ($C_1$)

Order 280: $C_2\times D_{70}$ ($C_2$), $C_{14}\times D_{10}$ ($C_2$), $C_{14}:C_{20}$ ($C_2$), $C_7:D_{20}$ ($C_2$)

Order 140: $C_7\times D_{10}$ ($C_2^2$), $C_7:C_{20}$ ($C_2^2$), $C_2\times C_{70}$ ($C_2^2$), $D_{70}$ ($C_2^2$)

Order 70: $C_{70}$ ($C_2^3$), $C_{70}$ ($D_4$)

Order 56: $C_{14}:C_4$ ($D_5$)

Order 40: $C_2\times D_{10}$ ($D_7$)

Order 35: $C_{35}$ ($C_2\times D_4$)

Order 28: $C_2\times C_{14}$ ($D_{10}$), $C_7:C_4$ ($D_{10}$)

Order 20: $C_2\times C_{10}$ ($D_{14}$), $D_{10}$ ($D_{14}$)

Order 14: $C_{14}$ ($D_{20}$), $C_{14}$ ($C_2\times D_{10}$)

Order 10: $C_{10}$ ($C_7:D_4$), $C_{10}$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_2\times D_{20}$)

Order 5: $C_5$ ($C_{14}:D_4$)

Order 4: $C_2^2$ ($D_5\times D_7$)

Order 2: $C_2$ ($D_5\times D_{14}$), $C_2$ ($C_7:D_{20}$)

Order 1: $C_1$ ($C_{14}:D_{20}$)

Series

Derived series

$C_{14}:D_{20}$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_{14}:D_{20}$

$\rhd$

$C_{14}\times D_{10}$

$\rhd$

$C_2\times C_{70}$

$\rhd$

$C_{70}$

$\rhd$

$C_{14}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{14}:D_{20}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 42 larger groups in the database.

This group is a maximal quotient of 32 larger groups in the database.

Character theory

Complex character table

See the $74 \times 74$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $26 \times 26$ rational character table.