Abstract group 1120.1008: $C_{140}.C_2^3$

• Label: 1120.1008
• Order: \( 2^{5} \cdot 5 \cdot 7 \)
• Exponent: \( 2^{2} \cdot 5 \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \cdot 3^{2} \)
• Perm deg.: $20$
• Trans deg.: $280$
• Rank: $4$

Group information

Description:	$C_{140}.C_2^3$
Order:	\(1120\)\(\medspace = 2^{5} \cdot 5 \cdot 7 \)
Exponent:	\(140\)\(\medspace = 2^{2} \cdot 5 \cdot 7 \)
Automorphism group:	$C_2^2\times F_5\times S_4\times F_7$, of order \(80640\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \)
Composition factors:	$C_2$ x 5, $C_5$, $C_7$
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	95	288	4	6	60	66	192	216	24	24	144	1120
Conjugacy classes	1	7	12	2	3	6	9	12	18	6	6	18	100
Divisions	1	7	12	1	1	3	3	6	6	1	1	3	45
Autjugacy classes	1	4	4	1	1	2	2	2	2	1	1	1	22

Dimension	1	2	4	6	8	12	24	48
Irr. complex chars.	16	44	34	0	6	0	0	0	100
Irr. rational chars.	16	4	8	8	2	2	4	1	45

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$280$
Rank:	$4$
Inequivalent generating quadruples:	$5937120$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	16	96
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	$\langle a, b, c, d \mid b^{2}=c^{2}=d^{140}=[a,b]=[a,c]=[b,c]=1, a^{2}=d^{70}, d^{a}=d^{71}, d^{b}=d^{29}, d^{c}=d^{69} \rangle$
Permutation group:	Degree $20$ $\langle(6,7)(8,10)(9,11)(12,13)(15,16)(17,18)(19,20), (2,3)(4,5)(6,7)(8,10)(9,11) \!\cdots\! \rangle$
Matrix group:	$\left\langle \left(\begin{array}{rr} 76 & 60 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 71 & 0 \\ 0 & 71 \end{array}\right), \left(\begin{array}{rr} 1 & 21 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 64 & 0 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 15 \\ 0 & 1 \end{array}\right), \left(\begin{array}{rr} 1 & 35 \\ 35 & 71 \end{array}\right), \left(\begin{array}{rr} 64 & 70 \\ 70 & 29 \end{array}\right) \right\rangle \subseteq \GL_{2}(\Z/105\Z)$
Direct product:	$Q_8$ $\, \times\, $ $D_5$ $\, \times\, $ $D_7$
Semidirect product:	$(D_{35}:Q_8)$ $\,\rtimes\,$ $C_2$	$(Q_8\times D_{35})$ $\,\rtimes\,$ $C_2$	$D_{35}$ $\,\rtimes\,$ $(C_2\times Q_8)$	$(C_7:Q_8)$ $\,\rtimes\,$ $D_{10}$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$D_{70}$ . $C_2^3$	$C_{70}$ . $C_2^4$	$C_{140}$ . $C_2^3$	$(C_4\times D_7)$ . $D_{10}$	all 28

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{5}$
Commutator length:	$1$

Subgroups

There are 2596 subgroups in 312 conjugacy classes, 122 normal (24 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $D_{10}\times D_{14}$
Commutator:	$G' \simeq$ $C_{70}$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $D_{10}\times D_{14}$
Fitting:	$\operatorname{Fit} \simeq$ $Q_8\times C_{35}$	$G/\operatorname{Fit} \simeq$ $C_2^2$
Radical:	$R \simeq$ $C_{140}.C_2^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_{70}$	$G/\operatorname{soc} \simeq$ $C_2^4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times Q_8$
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

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 1120: $C_{140}.C_2^3$

Order 560: $C_{20}:D_{14}$ x 3, $D_{35}:Q_8$ x 3, $C_{20}.D_{14}$ x 3, $C_{20}.D_{14}$ x 3, $Q_8\times D_{35}$, $C_{28}.D_{10}$, $C_{20}.D_{14}$

Order 280: $C_{35}:Q_8$ x 6, $D_{35}:C_4$ x 3, $C_{10}.D_{14}$ x 3, $C_{10}.D_{14}$ x 3, $C_4\times D_{35}$ x 3, $C_{35}:Q_8$ x 3, $D_5\times C_{28}$ x 3, $C_{35}:Q_8$ x 3, $D_7\times C_{20}$ x 3, $C_{35}:Q_8$ x 3, $D_5\times D_{14}$, $Q_8\times C_{35}$

Order 224: $Q_8\times D_{14}$

Order 160: $Q_8\times D_{10}$

Order 140: $D_5\times D_7$ x 4, $C_{140}$ x 3, $C_{35}:C_4$ x 3, $C_7:C_{20}$ x 3, $C_5:C_{28}$ x 3, $C_7\times D_{10}$, $C_5\times D_{14}$, $D_{70}$

Order 112: $Q_8\times D_7$ x 8, $C_4\times D_{14}$ x 3, $C_{14}:Q_8$ x 3, $Q_8\times C_{14}$

Order 80: $Q_8\times D_5$ x 8, $C_4\times D_{10}$ x 3, $C_{10}:Q_8$ x 3, $Q_8\times C_{10}$

Order 70: $D_{35}$ x 2, $C_5\times D_7$ x 2, $C_7\times D_5$ x 2, $C_{70}$

Order 56: $C_4\times D_7$ x 12, $C_7:Q_8$ x 12, $C_7\times Q_8$ x 4, $C_2\times C_{28}$ x 3, $C_{14}:C_4$ x 3, $C_2\times D_{14}$

Order 40: $C_4\times D_5$ x 12, $C_5:Q_8$ x 12, $C_5\times Q_8$ x 4, $C_2\times C_{20}$ x 3, $C_{10}:C_4$ x 3, $C_2\times D_{10}$

Order 35: $C_{35}$

Order 32: $C_2^2\times Q_8$

Order 28: $D_{14}$ x 6, $C_{28}$ x 6, $C_7:C_4$ x 6, $C_2\times C_{14}$

Order 20: $D_{10}$ x 6, $C_{20}$ x 6, $C_5:C_4$ x 6, $C_2\times C_{10}$

Order 16: $C_2\times Q_8$ x 12, $C_2^2\times C_4$ x 3

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

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

Order 8: $C_2\times C_4$ x 18, $Q_8$ x 16, $C_2^3$

Order 7: $C_7$

Order 5: $C_5$

Order 4: $C_4$ x 12, $C_2^2$ x 7

Order 2: $C_2$ x 7

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 1120: $C_{140}.C_2^3$

Order 560: $Q_8\times D_{35}$, $C_{28}.D_{10}$, $C_{20}.D_{14}$, $C_{20}:D_{14}$, $D_{35}:Q_8$, $C_{20}.D_{14}$, $C_{20}.D_{14}$

Order 280: $D_{35}:C_4$, $C_{10}.D_{14}$, $C_{10}.D_{14}$, $D_5\times D_{14}$, $Q_8\times C_{35}$, $C_4\times D_{35}$, $C_{35}:Q_8$, $D_5\times C_{28}$, $C_{35}:Q_8$, $D_7\times C_{20}$, $C_{35}:Q_8$, $C_{35}:Q_8$

Order 224: $Q_8\times D_{14}$

Order 160: $Q_8\times D_{10}$

Order 140: $C_7\times D_{10}$, $C_5\times D_{14}$, $D_5\times D_7$, $C_{140}$, $C_{35}:C_4$, $C_7:C_{20}$, $D_{70}$, $C_5:C_{28}$

Order 112: $Q_8\times D_7$ x 4, $Q_8\times C_{14}$, $C_4\times D_{14}$, $C_{14}:Q_8$

Order 80: $Q_8\times D_5$ x 4, $Q_8\times C_{10}$, $C_4\times D_{10}$, $C_{10}:Q_8$

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

Order 56: $C_4\times D_7$ x 4, $C_7:Q_8$ x 3, $C_7\times Q_8$ x 2, $C_2\times C_{28}$, $C_{14}:C_4$, $C_2\times D_{14}$

Order 40: $C_4\times D_5$ x 4, $C_5:Q_8$ x 3, $C_5\times Q_8$ x 2, $C_2\times C_{20}$, $C_{10}:C_4$, $C_2\times D_{10}$

Order 35: $C_{35}$

Order 32: $C_2^2\times Q_8$

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

Order 20: $D_{10}$ x 3, $C_{20}$ x 2, $C_5:C_4$ x 2, $C_2\times C_{10}$

Order 16: $C_2\times Q_8$ x 6, $C_2^2\times C_4$

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

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

Order 8: $C_2\times C_4$ x 6, $Q_8$ x 5, $C_2^3$

Order 7: $C_7$

Order 5: $C_5$

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

Order 2: $C_2$ x 4

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 1120: $C_{140}.C_2^3$ ($C_1$)

Order 560: $C_{20}:D_{14}$ ($C_2$) x 3, $D_{35}:Q_8$ ($C_2$) x 3, $C_{20}.D_{14}$ ($C_2$) x 3, $C_{20}.D_{14}$ ($C_2$) x 3, $Q_8\times D_{35}$ ($C_2$), $C_{28}.D_{10}$ ($C_2$), $C_{20}.D_{14}$ ($C_2$)

Order 280: $C_{35}:Q_8$ ($C_2^2$) x 6, $D_{35}:C_4$ ($C_2^2$) x 3, $C_{10}.D_{14}$ ($C_2^2$) x 3, $C_{10}.D_{14}$ ($C_2^2$) x 3, $C_4\times D_{35}$ ($C_2^2$) x 3, $C_{35}:Q_8$ ($C_2^2$) x 3, $D_5\times C_{28}$ ($C_2^2$) x 3, $C_{35}:Q_8$ ($C_2^2$) x 3, $D_7\times C_{20}$ ($C_2^2$) x 3, $C_{35}:Q_8$ ($C_2^2$) x 3, $D_5\times D_{14}$ ($C_2^2$), $Q_8\times C_{35}$ ($C_2^2$)

Order 140: $D_5\times D_7$ ($Q_8$) x 4, $C_{140}$ ($C_2^3$) x 3, $C_{35}:C_4$ ($C_2^3$) x 3, $C_7:C_{20}$ ($C_2^3$) x 3, $C_5:C_{28}$ ($C_2^3$) x 3, $C_7\times D_{10}$ ($C_2^3$), $C_5\times D_{14}$ ($C_2^3$), $D_{70}$ ($C_2^3$)

Order 112: $Q_8\times D_7$ ($D_5$)

Order 80: $Q_8\times D_5$ ($D_7$)

Order 70: $D_{35}$ ($C_2\times Q_8$) x 2, $C_5\times D_7$ ($C_2\times Q_8$) x 2, $C_7\times D_5$ ($C_2\times Q_8$) x 2, $C_{70}$ ($C_2^4$)

Order 56: $C_4\times D_7$ ($D_{10}$) x 3, $C_7:Q_8$ ($D_{10}$) x 3, $C_7\times Q_8$ ($D_{10}$)

Order 40: $C_4\times D_5$ ($D_{14}$) x 3, $C_5:Q_8$ ($D_{14}$) x 3, $C_5\times Q_8$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2^2\times Q_8$)

Order 28: $C_{28}$ ($C_2\times D_{10}$) x 3, $C_7:C_4$ ($C_2\times D_{10}$) x 3, $D_{14}$ ($C_2\times D_{10}$)

Order 20: $C_{20}$ ($C_2\times D_{14}$) x 3, $C_5:C_4$ ($C_2\times D_{14}$) x 3, $D_{10}$ ($C_2\times D_{14}$)

Order 14: $D_7$ ($Q_8\times D_5$) x 2, $C_{14}$ ($C_2^2\times D_{10}$)

Order 10: $D_5$ ($Q_8\times D_7$) x 2, $C_{10}$ ($C_2^2\times D_{14}$)

Order 8: $Q_8$ ($D_5\times D_7$)

Order 7: $C_7$ ($Q_8\times D_{10}$)

Order 5: $C_5$ ($Q_8\times D_{14}$)

Order 4: $C_4$ ($D_5\times D_{14}$) x 3

Order 2: $C_2$ ($D_{10}\times D_{14}$)

Order 1: $C_1$ ($C_{140}.C_2^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 1120: $C_{140}.C_2^3$ ($C_1$)

Order 560: $Q_8\times D_{35}$ ($C_2$), $C_{28}.D_{10}$ ($C_2$), $C_{20}.D_{14}$ ($C_2$), $C_{20}:D_{14}$ ($C_2$), $D_{35}:Q_8$ ($C_2$), $C_{20}.D_{14}$ ($C_2$), $C_{20}.D_{14}$ ($C_2$)

Order 280: $D_{35}:C_4$ ($C_2^2$), $C_{10}.D_{14}$ ($C_2^2$), $C_{10}.D_{14}$ ($C_2^2$), $D_5\times D_{14}$ ($C_2^2$), $Q_8\times C_{35}$ ($C_2^2$), $C_4\times D_{35}$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$), $D_5\times C_{28}$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$), $D_7\times C_{20}$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$), $C_{35}:Q_8$ ($C_2^2$)

Order 140: $C_7\times D_{10}$ ($C_2^3$), $C_5\times D_{14}$ ($C_2^3$), $D_5\times D_7$ ($Q_8$), $C_{140}$ ($C_2^3$), $C_{35}:C_4$ ($C_2^3$), $C_7:C_{20}$ ($C_2^3$), $D_{70}$ ($C_2^3$), $C_5:C_{28}$ ($C_2^3$)

Order 112: $Q_8\times D_7$ ($D_5$)

Order 80: $Q_8\times D_5$ ($D_7$)

Order 70: $C_{70}$ ($C_2^4$), $D_{35}$ ($C_2\times Q_8$), $C_5\times D_7$ ($C_2\times Q_8$), $C_7\times D_5$ ($C_2\times Q_8$)

Order 56: $C_4\times D_7$ ($D_{10}$), $C_7:Q_8$ ($D_{10}$), $C_7\times Q_8$ ($D_{10}$)

Order 40: $C_4\times D_5$ ($D_{14}$), $C_5:Q_8$ ($D_{14}$), $C_5\times Q_8$ ($D_{14}$)

Order 35: $C_{35}$ ($C_2^2\times Q_8$)

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

Order 20: $D_{10}$ ($C_2\times D_{14}$), $C_{20}$ ($C_2\times D_{14}$), $C_5:C_4$ ($C_2\times D_{14}$)

Order 14: $C_{14}$ ($C_2^2\times D_{10}$), $D_7$ ($Q_8\times D_5$)

Order 10: $C_{10}$ ($C_2^2\times D_{14}$), $D_5$ ($Q_8\times D_7$)

Order 8: $Q_8$ ($D_5\times D_7$)

Order 7: $C_7$ ($Q_8\times D_{10}$)

Order 5: $C_5$ ($Q_8\times D_{14}$)

Order 4: $C_4$ ($D_5\times D_{14}$)

Order 2: $C_2$ ($D_{10}\times D_{14}$)

Order 1: $C_1$ ($C_{140}.C_2^3$)

Series

Derived series

$C_{140}.C_2^3$

$\rhd$

$C_{70}$

$\rhd$

$C_1$

Chief series

$C_{140}.C_2^3$

$\rhd$

$C_{20}:D_{14}$

$\rhd$

$D_5\times C_{28}$

$\rhd$

$C_{140}$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

$\rhd$

$C_7$

$\rhd$

$C_1$

Lower central series

$C_{140}.C_2^3$

$\rhd$

$C_{70}$

$\rhd$

$C_{35}$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$Q_8$

Character theory

Complex character table

See the $100 \times 100$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $45 \times 45$ rational character table.