Abstract group 560.176: $D_{10}\times D_{14}$

• Label: 560.176
• Order: \( 2^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2 \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{6} \cdot 3^{2} \)
• Perm deg.: $16$
• Trans deg.: $140$
• Rank: $4$

Group information

Description:	$D_{10}\times D_{14}$
Order:	\(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \)
Exponent:	\(70\)\(\medspace = 2 \cdot 5 \cdot 7 \)
Automorphism group:	$F_5\times C_2^2:S_4\times F_7$, of order \(80640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 4, $C_5$, $C_7$
Derived length:	$2$

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

Group statistics

Order	1	2	5	7	10	14	35	70
Elements	1	191	4	6	124	138	24	72	560
Conjugacy classes	1	15	2	3	14	21	6	18	80
Divisions	1	15	1	1	7	7	1	3	36
Autjugacy classes	1	4	1	1	2	2	1	1	13

Dimension	1	2	4	6	24
Irr. complex chars.	16	40	24	0	0	80
Irr. rational chars.	16	0	8	8	4	36

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$140$
Rank:	$4$
Inequivalent generating quadruples:	$371070$

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:	$\langle a, b, c, d \mid a^{2}=b^{2}=c^{2}=d^{70}=[a,b]=[a,c]=[a,d]=[b,c]=1, d^{b}=d^{29}, d^{c}=d^{41} \rangle$
Permutation group:	Degree $16$ $\langle(11,12)(13,14)(15,16), (2,3)(4,5), (6,7)(8,9), (6,8)(7,9), (1,2,4,5,3), (10,11,13,15,16,14,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 5 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 7 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 29 & 0 \\ 0 & 29 \end{array}\right), \left(\begin{array}{rr} 6 & 0 \\ 0 & 6 \end{array}\right), \left(\begin{array}{rr} 1 & 30 \\ 0 & 6 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/35\Z)$
Direct product:	$C_2$ ${}^2$ $\, \times\, $ $D_5$ $\, \times\, $ $D_7$
Semidirect product:	$D_{70}$ $\,\rtimes\,$ $C_2^2$	$D_{35}$ $\,\rtimes\,$ $C_2^3$	$C_{70}$ $\,\rtimes\,$ $C_2^3$	$C_{35}$ $\,\rtimes\,$ $C_2^4$	all 18
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_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 2248 subgroups in 268 conjugacy classes, 104 normal (14 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:	$G' \simeq$ $C_{35}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $D_{10}\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{70}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $D_{10}\times D_{14}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_{70}$	$G/\operatorname{soc} \simeq$ $C_2^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4$
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

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

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.

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

Subgroup information

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

Classes of subgroups up to conjugation

Order 560: $D_{10}\times D_{14}$

Order 280: $D_5\times D_{14}$ x 12, $C_2\times D_{70}$, $C_{14}\times D_{10}$, $C_{10}\times D_{14}$

Order 140: $D_5\times D_7$ x 16, $C_7\times D_{10}$ x 6, $C_5\times D_{14}$ x 6, $D_{70}$ x 6, $C_2\times C_{70}$

Order 112: $C_2^2\times D_{14}$

Order 80: $C_2^2\times D_{10}$

Order 70: $D_{35}$ x 4, $C_5\times D_7$ x 4, $C_7\times D_5$ x 4, $C_{70}$ x 3

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

Order 40: $C_2\times D_{10}$ x 14, $C_2^2\times C_{10}$

Order 35: $C_{35}$

Order 28: $D_{14}$ x 28, $C_2\times C_{14}$ x 7

Order 20: $D_{10}$ x 28, $C_2\times C_{10}$ x 7

Order 16: $C_2^4$

Order 14: $D_7$ x 8, $C_{14}$ x 7

Order 10: $D_5$ x 8, $C_{10}$ x 7

Order 8: $C_2^3$ x 15

Order 7: $C_7$

Order 5: $C_5$

Order 4: $C_2^2$ x 35

Order 2: $C_2$ x 15

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 560: $D_{10}\times D_{14}$

Order 280: $C_2\times D_{70}$, $C_{14}\times D_{10}$, $C_{10}\times D_{14}$, $D_5\times D_{14}$

Order 140: $C_7\times D_{10}$, $C_5\times D_{14}$, $D_5\times D_7$, $C_2\times C_{70}$, $D_{70}$

Order 112: $C_2^2\times D_{14}$

Order 80: $C_2^2\times D_{10}$

Order 70: $C_{70}$, $D_{35}$, $C_5\times D_7$, $C_7\times D_5$

Order 56: $C_2\times D_{14}$ x 3, $C_2^2\times C_{14}$

Order 40: $C_2\times D_{10}$ x 3, $C_2^2\times C_{10}$

Order 35: $C_{35}$

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

Order 20: $D_{10}$ x 3, $C_2\times C_{10}$ x 2

Order 16: $C_2^4$

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

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

Order 8: $C_2^3$ x 4

Order 7: $C_7$

Order 5: $C_5$

Order 4: $C_2^2$ x 5

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 560: $D_{10}\times D_{14}$ ($C_1$)

Order 280: $D_5\times D_{14}$ ($C_2$) x 12, $C_2\times D_{70}$ ($C_2$), $C_{14}\times D_{10}$ ($C_2$), $C_{10}\times D_{14}$ ($C_2$)

Order 140: $D_5\times D_7$ ($C_2^2$) x 16, $C_7\times D_{10}$ ($C_2^2$) x 6, $C_5\times D_{14}$ ($C_2^2$) x 6, $D_{70}$ ($C_2^2$) x 6, $C_2\times C_{70}$ ($C_2^2$)

Order 70: $D_{35}$ ($C_2^3$) x 4, $C_5\times D_7$ ($C_2^3$) x 4, $C_7\times D_5$ ($C_2^3$) x 4, $C_{70}$ ($C_2^3$) x 3

Order 56: $C_2\times D_{14}$ ($D_5$)

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

Order 35: $C_{35}$ ($C_2^4$)

Order 28: $D_{14}$ ($D_{10}$) x 6, $C_2\times C_{14}$ ($D_{10}$)

Order 20: $D_{10}$ ($D_{14}$) x 6, $C_2\times C_{10}$ ($D_{14}$)

Order 14: $D_7$ ($C_2\times D_{10}$) x 4, $C_{14}$ ($C_2\times D_{10}$) x 3

Order 10: $D_5$ ($C_2\times D_{14}$) x 4, $C_{10}$ ($C_2\times D_{14}$) x 3

Order 7: $C_7$ ($C_2^2\times D_{10}$)

Order 5: $C_5$ ($C_2^2\times D_{14}$)

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

Order 2: $C_2$ ($D_5\times D_{14}$) x 3

Order 1: $C_1$ ($D_{10}\times D_{14}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 560: $D_{10}\times D_{14}$ ($C_1$)

Order 280: $C_2\times D_{70}$ ($C_2$), $C_{14}\times D_{10}$ ($C_2$), $C_{10}\times D_{14}$ ($C_2$), $D_5\times D_{14}$ ($C_2$)

Order 140: $C_7\times D_{10}$ ($C_2^2$), $C_5\times D_{14}$ ($C_2^2$), $D_5\times D_7$ ($C_2^2$), $C_2\times C_{70}$ ($C_2^2$), $D_{70}$ ($C_2^2$)

Order 70: $C_{70}$ ($C_2^3$), $D_{35}$ ($C_2^3$), $C_5\times D_7$ ($C_2^3$), $C_7\times D_5$ ($C_2^3$)

Order 56: $C_2\times D_{14}$ ($D_5$)

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

Order 35: $C_{35}$ ($C_2^4$)

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

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

Order 14: $C_{14}$ ($C_2\times D_{10}$), $D_7$ ($C_2\times D_{10}$)

Order 10: $C_{10}$ ($C_2\times D_{14}$), $D_5$ ($C_2\times D_{14}$)

Order 7: $C_7$ ($C_2^2\times D_{10}$)

Order 5: $C_5$ ($C_2^2\times D_{14}$)

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

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

Order 1: $C_1$ ($D_{10}\times D_{14}$)

Series

Derived series

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

$\rhd$

$C_{35}$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$D_5\times D_{14}$

$\rhd$

$C_7\times D_{10}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 27 larger groups in the database.

This group is a maximal quotient of 59 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 $36 \times 36$ rational character table.