Abstract group 4970.b: $C_{142}:C_{35}$

• Label: 4970.b
• Order: \( 2 \cdot 5 \cdot 7 \cdot 71 \)
• Exponent: \( 2 \cdot 5 \cdot 7 \cdot 71 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 5 \cdot 7 \)
• $\card{Z(G)}$: \( 2 \cdot 5 \)
• $\card{\Aut(G)}$: \( 2^{3} \cdot 5 \cdot 7 \cdot 71 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 5 \)
• Perm deg.: $78$
• Trans deg.: $710$
• Rank: $2$

Group information

Description:	$C_{142}:C_{35}$
Order:	\(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \)
Exponent:	\(4970\)\(\medspace = 2 \cdot 5 \cdot 7 \cdot 71 \)
Automorphism group:	$C_{71}:(C_2\times C_{140})$, of order \(19880\)\(\medspace = 2^{3} \cdot 5 \cdot 7 \cdot 71 \)
Composition factors:	$C_2$, $C_5$, $C_7$, $C_{71}$
Derived length:	$2$

This group is nonabelian, a Z-group (hence solvable, supersolvable, monomial, metacyclic, metabelian, and an A-group), and hyperelementary for $p = 7$.

Group statistics

Order	1	2	5	7	10	14	35	70	71	142	355	710
Elements	1	1	4	426	4	426	1704	1704	70	70	280	280	4970
Conjugacy classes	1	1	4	6	4	6	24	24	10	10	40	40	170
Divisions	1	1	1	1	1	1	1	1	1	1	1	1	12
Autjugacy classes	1	1	1	6	1	6	6	6	1	1	1	1	32

Dimension	1	4	6	7	24	70	280
Irr. complex chars.	70	0	0	100	0	0	0	170
Irr. rational chars.	2	2	2	0	2	2	2	12

Minimal presentations

Permutation degree:	$78$
Transitive degree:	$710$
Rank:	$2$
Inequivalent generating pairs:	$864$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	7	14	280
Arbitrary	7	14	74

Constructions

Presentation:	$\langle a, b \mid a^{7}=b^{710}=1, b^{a}=b^{471} \rangle$
Permutation group:	Degree $78$ $\langle(77,78), (72,73,74,75,76), (2,3,5,9,13,22,36)(4,7,8,14,24,39,57)(6,11,19,31,48,29,45) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 1 & 1 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 49 & 0 \\ 0 & 4 \end{array}\right), \left(\begin{array}{rr} 51 & 0 \\ 0 & 39 \end{array}\right) \right\rangle \subseteq \GL_{2}(\F_{71})$
Direct product:	$C_2$ $\, \times\, $ $C_5$ $\, \times\, $ $(C_{71}:C_7)$
Semidirect product:	$C_{710}$ $\,\rtimes\,$ $C_7$	$C_{71}$ $\,\rtimes\,$ $C_{70}$	$C_{355}$ $\,\rtimes\,$ $C_{14}$	$C_{142}$ $\,\rtimes\,$ $C_{35}$	more information
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:	$C_{70} \simeq C_{2} \times C_{5} \times C_{7}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 296 subgroups in 16 conjugacy classes, 12 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_{10}$	$G/Z \simeq$ $C_{71}:C_7$
Commutator:	$G' \simeq$ $C_{71}$	$G/G' \simeq$ $C_{70}$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_{142}:C_{35}$
Fitting:	$\operatorname{Fit} \simeq$ $C_{710}$	$G/\operatorname{Fit} \simeq$ $C_7$
Radical:	$R \simeq$ $C_{142}:C_{35}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{710}$	$G/\operatorname{soc} \simeq$ $C_7$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$
71-Sylow subgroup:	$P_{ 71 } \simeq$ $C_{71}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

Classes of subgroups up to automorphism

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 4970: $C_{142}:C_{35}$

Order 2485: $C_{355}:C_7$

Order 994: $C_{142}:C_7$

Order 710: $C_{710}$

Order 497: $C_{71}:C_7$

Order 355: $C_{355}$

Order 142: $C_{142}$

Order 71: $C_{71}$

Order 70: $C_{70}$

Order 35: $C_{35}$

Order 14: $C_{14}$

Order 10: $C_{10}$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4970: $C_{142}:C_{35}$

Order 2485: $C_{355}:C_7$

Order 994: $C_{142}:C_7$

Order 710: $C_{710}$

Order 497: $C_{71}:C_7$

Order 355: $C_{355}$

Order 142: $C_{142}$

Order 71: $C_{71}$

Order 70: $C_{70}$

Order 35: $C_{35}$

Order 14: $C_{14}$

Order 10: $C_{10}$

Order 7: $C_7$

Order 5: $C_5$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4970: $C_{142}:C_{35}$ ($C_1$)

Order 2485: $C_{355}:C_7$ ($C_2$)

Order 994: $C_{142}:C_7$ ($C_5$)

Order 710: $C_{710}$ ($C_7$)

Order 497: $C_{71}:C_7$ ($C_{10}$)

Order 355: $C_{355}$ ($C_{14}$)

Order 142: $C_{142}$ ($C_{35}$)

Order 71: $C_{71}$ ($C_{70}$)

Order 10: $C_{10}$ ($C_{71}:C_7$)

Order 5: $C_5$ ($C_{142}:C_7$)

Order 2: $C_2$ ($C_{355}:C_7$)

Order 1: $C_1$ ($C_{142}:C_{35}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4970: $C_{142}:C_{35}$ ($C_1$)

Order 2485: $C_{355}:C_7$ ($C_2$)

Order 994: $C_{142}:C_7$ ($C_5$)

Order 710: $C_{710}$ ($C_7$)

Order 497: $C_{71}:C_7$ ($C_{10}$)

Order 355: $C_{355}$ ($C_{14}$)

Order 142: $C_{142}$ ($C_{35}$)

Order 71: $C_{71}$ ($C_{70}$)

Order 10: $C_{10}$ ($C_{71}:C_7$)

Order 5: $C_5$ ($C_{142}:C_7$)

Order 2: $C_2$ ($C_{355}:C_7$)

Order 1: $C_1$ ($C_{142}:C_{35}$)

Series

Derived series	$C_{142}:C_{35}$	$\rhd$	$C_{71}$	$\rhd$	$C_1$
Chief series	$C_{142}:C_{35}$	$\rhd$	$C_{710}$	$\rhd$	$C_{355}$	$\rhd$	$C_{71}$	$\rhd$	$C_1$
Lower central series	$C_{142}:C_{35}$	$\rhd$	$C_{71}$
Upper central series	$C_1$	$\lhd$	$C_{10}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 1 larger groups in the database.

Character theory

Complex character table

See the $170 \times 170$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

		1A	2A	5A	7A	10A	14A	35A	70A	71A	142A	355A	710A
	Size	1	1	4	426	4	426	1704	1704	70	70	280	280
	2 P	1A	1A	5A	7A	5A	7A	35A	35A	71A	71A	355A	355A
	5 P	1A	2A	5A	7A	10A	14A	35A	70A	71A	142A	355A	710A
	7 P	1A	2A	1A	7A	2A	14A	7A	14A	71A	142A	71A	142A
	71 P	1A	2A	5A	1A	10A	2A	5A	10A	71A	142A	355A	710A
4970.2.1a		$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
4970.2.1b		$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
4970.2.1c		$4$	$4$	$-1$	$4$	$-1$	$4$	$-1$	$-1$	$4$	$4$	$-1$	$-1$
4970.2.1d		$4$	$-4$	$-1$	$4$	$1$	$-4$	$-1$	$1$	$4$	$-4$	$-1$	$1$
4970.2.1e		$6$	$6$	$6$	$-1$	$6$	$-1$	$-1$	$-1$	$6$	$6$	$6$	$6$
4970.2.1f		$6$	$-6$	$6$	$-1$	$-6$	$1$	$-1$	$1$	$6$	$-6$	$6$	$-6$
4970.2.1g		$24$	$24$	$-6$	$-4$	$-6$	$-4$	$1$	$1$	$24$	$24$	$-6$	$-6$
4970.2.1h		$24$	$-24$	$-6$	$-4$	$6$	$4$	$1$	$-1$	$24$	$-24$	$-6$	$6$
4970.2.7a		$70$	$70$	$70$	$0$	$70$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
4970.2.7b		$70$	$-70$	$70$	$0$	$-70$	$0$	$0$	$0$	$-1$	$1$	$-1$	$1$
4970.2.7c		$280$	$280$	$-70$	$0$	$-70$	$0$	$0$	$0$	$-4$	$-4$	$1$	$1$
4970.2.7d		$280$	$-280$	$-70$	$0$	$70$	$0$	$0$	$0$	$-4$	$4$	$1$	$-1$