Abstract group 560.158: $C_2\times D_{140}$

• Label: 560.158
• Order: \( 2^{4} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \cdot 3 \)
• Perm deg.: $18$
• Trans deg.: $280$
• Rank: $3$

Group information

Description:	$C_2\times D_{140}$
Order:	\(560\)\(\medspace = 2^{4} \cdot 5 \cdot 7 \)
Exponent:	\(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$C_{70}.(C_2^3\times C_6).C_2^4$, of order \(53760\)\(\medspace = 2^{9} \cdot 3 \cdot 5 \cdot 7 \)
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	28	35	70	140
Elements	1	283	4	4	6	12	18	16	24	24	72	96	560
Conjugacy classes	1	7	2	2	3	6	9	8	12	12	36	48	146
Divisions	1	7	2	1	1	3	3	2	2	1	3	2	28
Autjugacy classes	1	3	1	1	1	2	2	1	1	1	2	1	17

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	8	138	0	0	0	0	0	0	146
Irr. rational chars.	8	2	4	4	2	2	4	2	28

Minimal presentations

Permutation degree:	$18$
Transitive degree:	$280$
Rank:	$3$
Inequivalent generating triples:	$1008$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	none	none
Arbitrary	3	3	12

Constructions

Presentation:	$\langle a, b, c \mid a^{2}=b^{2}=c^{140}=[a,b]=[b,c]=1, c^{a}=c^{139} \rangle$
Permutation group:	Degree $18$ $\langle(2,3)(4,5)(6,7)(8,9)(11,12)(13,14)(15,16)(17,18), (6,8)(7,9)(17,18), (6,9,8,7)(17,18), (6,8)(7,9), (1,2,4,5,3), (10,11,13,15,16,14,12)\rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 64 & 35 \\ 35 & 29 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 64 & 72 \\ 35 & 41 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 34 & 0 \\ 0 & 34 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$C_2$ $\, \times\, $ $D_{140}$
Semidirect product:	$C_{70}$ $\,\rtimes\,$ $D_4$ (2)	$C_4$ $\,\rtimes\,$ $D_{70}$ (2)	$C_{28}$ $\,\rtimes\,$ $D_{10}$ (2)	$C_{20}$ $\,\rtimes\,$ $D_{14}$ (2)	all 16
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^2$ . $D_{70}$	$C_{70}$ . $C_2^3$	$C_2$ . $(C_2\times D_{70})$	$(C_2\times C_{14})$ . $D_{10}$	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 1328 subgroups in 108 conjugacy classes, 43 normal (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $D_{70}$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2\times D_{70}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_{140}$	$G/\operatorname{Fit} \simeq$ $C_2$
Radical:	$R \simeq$ $C_2\times D_{140}$	$G/R \simeq$ $C_1$
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

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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 560: $C_2\times D_{140}$

Order 280: $D_{140}$ x 4, $C_2\times D_{70}$ x 2, $C_2\times C_{140}$

Order 140: $D_{70}$ x 8, $C_{140}$ x 2, $C_2\times C_{70}$

Order 112: $C_2\times D_{28}$

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

Order 70: $D_{35}$ x 4, $C_{70}$ x 3

Order 56: $D_{28}$ x 4, $C_2\times D_{14}$ x 2, $C_2\times C_{28}$

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

Order 35: $C_{35}$

Order 28: $D_{14}$ x 8, $C_{28}$ x 2, $C_2\times C_{14}$

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

Order 16: $C_2\times D_4$

Order 14: $D_7$ x 4, $C_{14}$ x 3

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: $C_2\times D_{140}$

Order 280: $C_2\times D_{70}$, $C_2\times C_{140}$, $D_{140}$

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

Order 112: $C_2\times D_{28}$

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

Order 70: $C_{70}$ x 2, $D_{35}$

Order 56: $C_2\times C_{28}$, $D_{28}$, $C_2\times D_{14}$

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

Order 35: $C_{35}$

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

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

Order 16: $C_2\times D_4$

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

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

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

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 3

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 560: $C_2\times D_{140}$ ($C_1$)

Order 280: $D_{140}$ ($C_2$) x 4, $C_2\times D_{70}$ ($C_2$) x 2, $C_2\times C_{140}$ ($C_2$)

Order 140: $D_{70}$ ($C_2^2$) x 4, $C_{140}$ ($C_2^2$) 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: $C_2\times C_{28}$ ($D_5$)

Order 40: $C_2\times C_{20}$ ($D_7$)

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

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

Order 20: $C_{20}$ ($D_{14}$) 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: $C_{10}$ ($D_{28}$) x 2, $C_{10}$ ($C_2\times D_{14}$)

Order 8: $C_2\times C_4$ ($D_{35}$)

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

Order 5: $C_5$ ($C_2\times D_{28}$)

Order 4: $C_4$ ($D_{70}$) x 2, $C_2^2$ ($D_{70}$)

Order 2: $C_2$ ($D_{140}$) x 2, $C_2$ ($C_2\times D_{70}$)

Order 1: $C_1$ ($C_2\times D_{140}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 560: $C_2\times D_{140}$ ($C_1$)

Order 280: $C_2\times D_{70}$ ($C_2$), $C_2\times C_{140}$ ($C_2$), $D_{140}$ ($C_2$)

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

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

Order 56: $C_2\times C_{28}$ ($D_5$)

Order 40: $C_2\times C_{20}$ ($D_7$)

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

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

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

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

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

Order 8: $C_2\times C_4$ ($D_{35}$)

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

Order 5: $C_5$ ($C_2\times D_{28}$)

Order 4: $C_2^2$ ($D_{70}$), $C_4$ ($D_{70}$)

Order 2: $C_2$ ($C_2\times D_{70}$), $C_2$ ($D_{140}$)

Order 1: $C_1$ ($C_2\times D_{140}$)

Series

Derived series

$C_2\times D_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_2\times D_{140}$

$\rhd$

$C_2\times D_{70}$

$\rhd$

$D_{70}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_2\times D_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_2^2$

$\lhd$

$C_2\times C_4$

Supergroups

This group is a maximal subgroup of 56 larger groups in the database.

This group is a maximal quotient of 22 larger groups in the database.

Character theory

Complex character table

See the $146 \times 146$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $28 \times 28$ rational character table.